Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells 0128157534, 9780128157534

Table of contents :
Title-page_2020_Fundamentals-of-Heat-and-Fluid-Flow-in-High-Temperature-Fuel
Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells
Copyright_2020_Fundamentals-of-Heat-and-Fluid-Flow-in-High-Temperature-Fuel-
Copyright
Dedication_2020_Fundamentals-of-Heat-and-Fluid-Flow-in-High-Temperature-Fuel
Dedication
Contents_2020_Fundamentals-of-Heat-and-Fluid-Flow-in-High-Temperature-Fuel-C
Contents
About-the-autho_2020_Fundamentals-of-Heat-and-Fluid-Flow-in-High-Temperature
About the authors
Preface_2020_Fundamentals-of-Heat-and-Fluid-Flow-in-High-Temperature-Fuel-Ce
Preface
Acknowledgmen_2020_Fundamentals-of-Heat-and-Fluid-Flow-in-High-Temperature-F
Acknowledgments
Chapter-1---Introduction-t_2020_Fundamentals-of-Heat-and-Fluid-Flow-in-High-
1 Introduction to fuel cells
1.1 What is a fuel cell?
1.2 How does a fuel cell work?
1.3 Types of fuel cells
1.3.1 Hydroxide ion exchange fuel cell
1.3.2 Oxide ion exchange fuel cell
1.3.3 Proton exchange fuel cell
1.3.4 Carbonate ion exchange fuel cell
1.4 Thermodynamics of fuel cells
References
Chapter-2---Classification-of-s_2020_Fundamentals-of-Heat-and-Fluid-Flow-in-
2 Classification of solid oxide fuel cells
2.1 Historical summary
2.2 Geometrical types
2.2.1 Planar design
2.2.2 Tubular design
2.2.3 High-power density design
2.2.4 Delta design
2.2.5 Button design
2.3 Cell types in terms of its support
2.3.1 Electrolyte-supported solid oxide fuel cell
2.3.2 Cathode-supported solid oxide fuel cell
2.3.3 Anode-supported solid oxide fuel cell
2.4 Solid oxide fuel cell classification based on flow patterns
2.5 Cell types in terms of its chamber number
2.5.1 Dual-chamber solid oxide fuel cell
2.5.2 Single-chamber solid oxide fuel cell
2.5.3 No-chamber solid oxide fuel cell
2.6 Single and stack cell designs
References
Chapter-3---Solid-oxide-fuel-ce_2020_Fundamentals-of-Heat-and-Fluid-Flow-in-
3 Solid oxide fuel cells in hybrid systems
3.1 Strategies for improving the efficiency of solid oxide fuel cell power generation systems
3.2 Thermodynamic cycle options in hybrid solid oxide fuel cell systems
3.3 Balance of plant equipment
3.3.1 Fuel desulfurization
3.3.2 Heat exchangers
3.3.3 Ejectors
3.3.4 Reformer
3.3.5 Afterburners
3.3.6 Power electronics
3.3.7 Other components
3.4 Basic solid oxide fuel cell/gas turbine hybrid cycle
3.5 Different configurations of solid oxide fuel cell hybrid systems
3.5.1 Direct thermal coupling scheme
3.5.2 Indirect thermal coupling scheme
3.5.3 Other types of coupling
3.6 Mathematical modeling of an solid oxide fuel cell/gas turbine hybrid system
References
Chapter-4---Fundamentals-of-_2020_Fundamentals-of-Heat-and-Fluid-Flow-in-Hig
4 Fundamentals of electrochemistry
4.1 The basic concepts of gas mixture category
4.1.1 Mass fractions and mole fractions
4.1.2 Ideal gas mixtures
4.1.3 Properties of gas mixtures
4.2 Conservation of species
4.3 Species source terms in solid oxide fuel cells
4.3.1 Chemical reactions
4.3.2 Electrochemical reactions
4.3.2.1 Electrochemical reaction rate
4.3.3 Some applicable boundary conditions for solid oxide fuel cells
4.3.3.1 Inflow boundary conditions
4.3.3.2 Outflow boundary condition
4.3.3.3 Insulation boundary conditions
4.3.3.4 Electrical potential boundary condition
4.3.3.5 Axial symmetry boundary condition
4.3.3.6 Continuity boundary condition
References
Further reading
Chapter-5---Fundamental-of-_2020_Fundamentals-of-Heat-and-Fluid-Flow-in-High
5 Fundamental of heat transfer
5.1 Different modes of heat transfer
5.1.1 Conduction heat transfer
5.1.2 Convection heat transfer
5.1.3 Radiation heat transfer
5.1.3.1 Schuster–Schwartzchild two-flux approximation
5.1.3.2 Rosseland approximation
5.2 Energy conservation
5.2.1 Heat equation in electrolytes
5.2.2 Heat equation in porous electrodes
5.2.3 Heat equation in channels
5.3 Solid oxide fuel cell’s source terms
5.3.1 Joule or Ohmic heat source
5.3.2 Irreversible heat source
5.3.3 Reversible heat sources
5.3.4 Heat source generated by chemical reactions
5.4 Some applicable boundary conditions for solid oxide fuel cells
5.4.1 Specified temperature
5.4.2 Thermal insulated boundary
5.4.3 Specified heat flux
5.4.4 Continuity
5.4.5 Outflow
5.4.6 Symmetry
5.4.7 Surface-to-ambient radiation
References
Chapter-6---Fundamentals-o_2020_Fundamentals-of-Heat-and-Fluid-Flow-in-High-
6 Fundamentals of fluid flow
6.1 Conservation of mass
6.1.1 Mass sources
6.1.1.1 Mass sources caused by chemical reactions
6.1.1.2 Mass sources caused by electrochemical reactions
6.2 Conservation of linear momentum
6.2.1 The Brinkman equation
6.2.2 The Navier–Stokes equations
6.2.3 Body (volume) force
6.3 Boundary conditions
6.3.1 Inlet boundary condition
6.3.2 Outlet boundary condition
6.3.3 Wall boundary condition
6.3.4 Axial symmetry
6.3.5 Continuity
References
Chapter-7---Case-st_2020_Fundamentals-of-Heat-and-Fluid-Flow-in-High-Tempera
7 Case studies
7.1 Case study 1: Stationary performance analysis of a dual chamber solid oxide fuel cell with hydrogen fuel
7.2 Case 2: Transient performance analysis of a dual chamber solid oxide fuel cell with hydrogen fuel
7.3 Case study 3: The effect of coplanar and perpendicular catalyst layer configurations on the performance of a single-cha...
References
Index_2020_Fundamentals-of-Heat-and-Fluid-Flow-in-High-Temperature-Fuel-Cell
Index

Citation preview

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

MAJID GHASSEMI Mechanical Engineering Faculty K. N. Toosi University of Technology, Tehran Iran

MAJID KAMVAR Department of Mechanical Engineering, Parand Branch, Islamic Azad University, Parand, Iran

ROBERT STEINBERGER-WILCKENS School of Chemical Engineering, University of Birmingham, Edgbaston, United Kingdom

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2020 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-12-815753-4 For Information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Katie Hammon Acquisitions Editor: Raquel Zanol Editorial Project Manager: Ali Afzal-Khan Production Project Manager: R.Vijay Bharath Cover Designer: Matthew Limbert Typeset by MPS Limited, Chennai, India

Dedication To my daughter, Fatimah, son, Alireza, and my wife, Monir Majid To my mother, Zahra, and father Hassan Majid

Contents About the authors Preface Acknowledgments

1. Introduction to fuel cells 1.1 What is a fuel cell? 1.2 How does a fuel cell work? 1.3 Types of fuel cells 1.3.1 Hydroxide ion exchange fuel cell 1.3.2 Oxide ion exchange fuel cell 1.3.3 Proton exchange fuel cell 1.3.4 Carbonate ion exchange fuel cell 1.4 Thermodynamics of fuel cells References

2. Classification of solid oxide fuel cells 2.1 Historical summary 2.2 Geometrical types 2.2.1 Planar design 2.2.2 Tubular design 2.2.3 High-power density design 2.2.4 Delta design 2.2.5 Button design 2.3 Cell types in terms of its support 2.3.1 Electrolyte-supported solid oxide fuel cell 2.3.2 Cathode-supported solid oxide fuel cell 2.3.3 Anode-supported solid oxide fuel cell 2.4 Solid oxide fuel cell classification based on flow patterns 2.5 Cell types in terms of its chamber number 2.5.1 Dual-chamber solid oxide fuel cell 2.5.2 Single-chamber solid oxide fuel cell 2.5.3 No-chamber solid oxide fuel cell 2.6 Single and stack cell designs References

xi xiii xv

1 1 2 3 3 4 6 7 8 15

17 18 21 21 22 23 24 25 27 28 28 30 32 36 36 37 41 42 45

vii

viii

Contents

3. Solid oxide fuel cells in hybrid systems 3.1 Strategies for improving the efficiency of solid oxide fuel cell power generation systems 3.2 Thermodynamic cycle options in hybrid solid oxide fuel cell systems 3.3 Balance of plant equipment 3.3.1 Fuel desulfurization 3.3.2 Heat exchangers 3.3.3 Ejectors 3.3.4 Reformer 3.3.5 Afterburners 3.3.6 Power electronics 3.3.7 Other components 3.4 Basic solid oxide fuel cell/gas turbine hybrid cycle 3.5 Different configurations of solid oxide fuel cell hybrid systems 3.5.1 Direct thermal coupling scheme 3.5.2 Indirect thermal coupling scheme 3.5.3 Other types of coupling 3.6 Mathematical modeling of an solid oxide fuel cell/gas turbine hybrid system References

4. Fundamentals of electrochemistry 4.1 The basic concepts of gas mixture category 4.1.1 Mass fractions and mole fractions 4.1.2 Ideal gas mixtures 4.1.3 Properties of gas mixtures 4.2 Conservation of species 4.3 Species source terms in solid oxide fuel cells 4.3.1 Chemical reactions 4.3.2 Electrochemical reactions 4.3.3 Some applicable boundary conditions for solid oxide fuel cells References Further reading

5. Fundamental of heat transfer 5.1 Different modes of heat transfer 5.1.1 Conduction heat transfer 5.1.2 Convection heat transfer 5.1.3 Radiation heat transfer 5.2 Energy conservation 5.2.1 Heat equation in electrolytes

47 48 49 51 51 52 53 54 55 55 56 56 56 58 61 63 65 73

75 76 76 77 79 80 87 88 91 96 98 99

101 103 103 107 109 113 115

Contents

5.2.2 Heat equation in porous electrodes 5.2.3 Heat equation in channels 5.3 Solid oxide fuel cell’s source terms 5.3.1 Joule or Ohmic heat source 5.3.2 Irreversible heat source 5.3.3 Reversible heat sources 5.3.4 Heat source generated by chemical reactions 5.4 Some applicable boundary conditions for solid oxide fuel cells 5.4.1 Specified temperature 5.4.2 Thermal insulated boundary 5.4.3 Specified heat flux 5.4.4 Continuity 5.4.5 Outflow 5.4.6 Symmetry 5.4.7 Surface-to-ambient radiation References

6. Fundamentals of fluid flow 6.1 Conservation of mass 6.1.1 Mass sources 6.2 Conservation of linear momentum 6.2.1 The Brinkman equation 6.2.2 The Navier Stokes equations 6.2.3 Body (volume) force 6.3 Boundary conditions 6.3.1 Inlet boundary condition 6.3.2 Outlet boundary condition 6.3.3 Wall boundary condition 6.3.4 Axial symmetry 6.3.5 Continuity References

7. Case studies 7.1 Case study 1: Stationary performance analysis of a dual chamber solid oxide fuel cell with hydrogen fuel 7.2 Case 2: Transient performance analysis of a dual chamber solid oxide fuel cell with hydrogen fuel 7.3 Case study 3: The effect of coplanar and perpendicular catalyst layer configurations on the performance of a single-chamber SOFC References Index

ix 116 117 118 118 118 118 119 120 120 120 121 121 121 122 123 124

125 126 127 132 132 133 134 134 134 136 137 138 139 139

141 142 153 160 171 175

About the authors Prof. Majid Ghassemi is Professor in the Department of Mechanical Engineering at the K. N. Toosi University of Technology; one of the most prestigious technical universities in Tehran, Iran. Professor Ghassemi has been recognized as Global Talent, also known as an Exceptional Talent, endorsed by the Royal Academy of Engineering of the United Kingdom, since 2015. He received that honor when he was acting as a Visiting Professor at the Centre for Fuel Cell and Hydrogen Research at the University of Birmingham, United Kingdom. He teaches graduate and undergraduate courses and conducts research in the area of heat transfer and its application in bio and micro sensors, drug delivery, fuel cells, micro channels and alternative energy. He has over 20 years of academic and industrial experience and served as the President of the K. N. Toosi University of Technology from 2010 to 2013. He has also served in several public and private boards and panels and supervised several undergraduate, Masters and PhD students, published several books and many journal and conference papers. He is currently Editor-in-Chief and Editorial Board member of many national and international journals as well as Organizing Committee Member of many international conferences. He also serves as board member in several international conferences. He received his PhD in Mechanical Engineering from Iowa State University in 1993.

Dr. Majid Kamvar is an Assistant Professor at the Department of Mechanical Engineering at the Parand Islamic Azad University (PIAU), Parand, Iran. Both his MSc and PhD work were performed in the area of Solid Oxide Fuel Cells (SOFCs) modeling. Dr. Kamvar’s research is focused on the numerical investigation of SOFCs behavior with the aim of SOFC performance enhancement and limitations overcoming. He is currently teaching graduate and undergraduate courses and conducts research in the areas of heat transfer as well as fluid flow and their applications in energy management of high temperature fuel cells. He has 10 years of academic and industrial experience and supervised several undergraduate, Masters, and PhD students. He has succeeded to publish

xi

xii

About the authors

several ISI journal papers and also attended in international conferences held in the area of fuel cells. Prof. Robert Steinberger-Wilckens is Professor for Fuel Cell and Hydrogen Research in the School of Chemical Engineering at the University of Birmingham, United Kingdom. He is director of both the Centre for Fuel Cell and Hydrogen Research and the Centre for Doctoral Training in Fuel Cells and their Fuels, which is led by the University of Birmingham, with participation by the universities of Nottingham, and Loughborough, Imperial College, and University College of London. He works and has worked in many areas across the fields of renewable energies, energy efficiency, fuel cells, hydrogen production and infrastructure, electric vehicle development, and environmental analysis for more than 25 years. During this period he has worked as a consultant, project engineer, research manager, and in academia, publishing over 250 scientific papers in journals and conference proceedings, as well as contributing book chapters. Over 100 students have been supervised by him in the preparation of their PhD, MRes, Diploma, and MSc theses. He has served on conference and editorial boards and is currently the Chair of the Scientific Committee of the European Fuel Cell and Hydrogen 2 Joint Undertaking.

Preface Fuel cells are energy conversion devices and one of the most efficient technologies of generating electricity. They are used in combined heat and power devices (CHP) or for vehicle propulsion, to name the two main applications. They are also employed in space flight, for military projects, in uninterrupted power supply systems, for portable power sources, and in waste water treatment, among others. Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells helps engineers who are interested in improving their understanding of heat, fluid, and mass transfer in high-temperature fuel cells, specifically Solid Oxide Fuel Cells (SOFCs), as well as those who want to pursue their career in related engineering fields. The aim of this book is to introduce the fundamentals of heat, fluid, and mass transfer and their applications in high-temperature fuel cells. The book briefly covers different types of fuel cells and discusses the SOFC in detail. Finally, the book introduces several case studies. The book • provides in-depth knowledge of fuel cells, especially SOFC, • provides broad coverage of important issues related to the heat transfer and fluid flow in high-temperature fuel cells, • explores applications of fuel cells in different industries, and • discusses case studies.

xiii

Acknowledgments We, the authors, would like to express our greatest respect to our parents, without whom none of this would have been possible. The authors would also like to extend their highest gratitude to their immediate family for providing endless support and encouragement during the entire endeavor. DR. Kamvar would like to appreciate the Fangenerator industrial group for its support to this project. We also appreciate assistance from our Masters and PhD students at the Nano and Fuel Cell Laboratory at the K.N. Toosi University of Technology. We gratefully acknowledge the peer reviewers’ committment and useful suggestions. Last and not least: We beg forgiveness of all those who have been with us over the course of the book and whose names we have failed to mention.

xv

CHAPTER 1

Introduction to fuel cells Contents 1.1 What is a fuel cell? 1.2 How does a fuel cell work? 1.3 Types of fuel cells 1.3.1 Hydroxide ion exchange fuel cell 1.3.2 Oxide ion exchange fuel cell 1.3.3 Proton exchange fuel cell 1.3.4 Carbonate ion exchange fuel cell 1.4 Thermodynamics of fuel cells References

1 2 3 3 4 6 7 8 15

The aim of this chapter is to introduce the fuel cell principle, a useful alternative to electric power generation based on conventional, fossil fuels. Using hydrogen, methane, or other simple hydrocarbons, such as methanol, ethanol, propane, or butane, the fuel cell system is categorized as green energy and an environmentally benign system because the output of these systems consists just of water (which does not harm the atmosphere or organisms), carbon dioxide (if carbon-containing fuels are used), heat, and electricity. Fuel cell systems are divided into different types depending on the nature of their electrolyte. In this chapter, different types of fuel cell are presented and their mechanisms discussed briefly. In addition, the basic thermodynamics of fuel cell operation are also described.

1.1 What is a fuel cell? A fuel cell, as an environmentally friendly device for providing electricity and heat, converts chemical energy available in chemical species directly to electrical energy by electrochemical reactions. The losses of the process are available in the form of heat. A fuel cell converts fuel, especially hydrogen, into water, heat, and electrical energy. Nowadays, the demand for conventional electric power
producing devices, such as internal Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells DOI: https://doi.org/10.1016/B978-0-12-815753-4.00001-4

© 2020 Elsevier Inc. All rights reserved.

1

2

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

combustion engines, is being significantly challenged because these traditional devices produce all three nitrogen oxides that are of particular harm to the environment. Consequently, fuel cell technology and development has been receiving increased attention. Historically, the first hydrogen fuel cell was suggested and successfully tested by Friedrich Schönbein and William Grove. The first successful fuel cell experiment was performed in Swansea, Wales. The setup consisted of two platinum electrodes which were placed in two test tubes filled with oxygen and hydrogen, respectively. These tubes were half submerged in a bath of dilute sulfuric acid as shown in Fig. 1.1.

1.2 How does a fuel cell work? The most basic fuel cell operation is the electrochemical reaction between hydrogen as a fuel and oxygen as an oxidant. In principle, a fuel cell consists of three main layers: anode (fuel) electrode, cathode (oxidant) electrode, and electrolyte, as shown in Fig. 1.2. In all types of fuel cells, the fuel mixture is directed into the anode electrode through the anode channel and the air mixture is directed to the cathode electrode through the cathode channel. Anode and cathode electrodes are porous where the fuel and air mixtures can diffuse and are conveyed to the electrode
electrolyte interfaces. The ion produced during the electrochemical reactions, which take place in the vicinity of the electrode
electrolyte interface, is

Figure 1.1 The first hydrogen fuel cell built by Grove in 1839.

Introduction to fuel cells

3

Figure 1.2 A schematic illustration of how a simplest hydrogen fuel cell work.

conducted through the electrolyte layer and the electrons are transferred through the external electric circuit, producing an electric current. Fig. 1.2 schematically depicts the procedure of the simplest type of hydrogen fuel cell.

1.3 Types of fuel cells Even though all fuel cells produce electricity directly from the chemical energy in the different fuels, especially hydrogen, the working principles of the different types of fuel cells are very different from each other. The main difference between them stems from the type of electrolyte (often confusingly called “membrane”) used. The nature of the electrolyte determines the nature of the ionic conductivity and thus the direction of the ionic current transferred between the electrodes. Based on the ion types produced via electrochemical reactions, the fuel cells are divided into four different categories: 1. hydroxide ion exchange membrane fuel cells; 2. oxide ion exchange membrane fuel cells; 3. carbonate ion exchange membrane fuel cells, all three with negatively charged ions ; and 4. proton exchange membrane fuel cells (PEMFCs), with positive charge of the ion.

1.3.1 Hydroxide ion exchange fuel cell In hydroxide ion exchange fuel cells, hydrogen is oxidized on the anode side and a hydroxide ion as well as an electron is produced, using a water molecule, on the cathode side. The hydroxide ion passes through the

4

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

membrane and reacts with the remaining proton from the hydrogen oxidation, producing water on the anode side. The electron released travels through the external electrical circuit and thus delivers a flow of electricity. Fig. 1.3 depicts the typical operation of a hydroxide ion exchange membrane fuel cell. The overall reactions are as follows: • Hydrogen oxidation • •

H2 1 2OH2 -2H2 O 1 2e2

(1.i)

O2 1 2H2 O 1 4e2 -4OH2

(1.ii)

Oxygen reduction

Overall reaction O2 1 2H2 -2H2 O

(1.iii)

The main example of a hydroxide ion exchange membrane type fuel cell is the alkaline fuel cell based on a potassium hydroxide electrolyte.

1.3.2 Oxide ion exchange fuel cell In oxide ion exchange fuel cells, a type of high-temperature fuel cell operating above 600°C, oxygen is reduced at the cathode, generating

Figure 1.3 Schematics of a hydroxide ion exchange fuel cell.

Introduction to fuel cells

5

oxide ions on the cathode side. At high temperatures, the fuel cell electrolyte behaves as an ionic conductor and an electronic insulator. The ions produced pass through the electrolyte layer and reach the anode catalyst layer to contribute to the electrochemical oxidation of hydrogen, producing water and electrons. Again, the electron is forced to pass the exterior circuit and deliver electric current. Fig. 1.4 depicts how an oxide ion exchange fuel cell operates. The electrochemical reactions in an oxide ion exchange electrolyte are as follows: • Hydrogen oxidation • •

H2 1 O22 -H2 O 1 2e2

(1.iv)

O2 1 4e2 -2O22

(1.v)

Oxygen reduction

Overall reaction O2 1 2H2 -2H2 O

(1.vi)

An example of oxide ion conducting fuel cells is the solid oxide fuel cell (SOFC). Due to the elevated temperature, high-temperature fuel cells can also utilize other fuels than hydrogen (see Sect. 1.3.4).

Figure 1.4 Schematic of an oxide ion exchange fuel cell.

6

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

1.3.3 Proton exchange fuel cell The proton exchange fuel cell operates at temperatures of 50°C
200°C. There are two types: the polymer electrolyte fuel cell (PEFC) and the phosphoric acid fuel cell (PAFC). In both fuel cell types, the proton ion resulting from hydrogen oxidation passes through the electrolyte from anode electrode toward the cathode (as shown in Fig. 1.5), forming water on the cathode side. In the PEFC, a special proton-conducting polymer is used as an actual membrane. Therefore, this type of fuel cell has given rise to the use of “membrane” as a synonym for “electrolyte” and it is also widely (but wrongly) known as “the” PEMFC. This overlooks the fact that a PAFC is also based on proton conductivity. Hydrogen flows through the anode electrode of the proton exchange fuel cell and is split into protons and electrons according to the reaction given in Eq. (1.vii). The protons generated pass through the membrane (electrolyte) and reach the cathode side where they react with oxygen ions resulting from oxygen reduction with the help of the electron flow, ultimately producing water. Like with all other types of fuel cell, the electrolyte, which is an electron insulator, forces the electron to pass through

Figure 1.5 Schematic of a proton exchange membrane fuel cell.

Introduction to fuel cells

7

the exterior circuit, supplying an electric flow (“electricity”). The electrochemical reactions taking place in a PEFC or PAFC are as follows: • Hydrogen oxidation • •

H2 -2H1 1 2e2

(1.vii)

O2 1 4H1 1 4e2 -2H2 O

(1.viii)

Oxygen reduction

Overall reaction O2 12H2 -2H2 O

(1.ix)

1.3.4 Carbonate ion exchange fuel cell The carbonate ion exchange membrane fuel cell is categorized as a type of high-temperature fuel cell. Its operating temperature, above 600°C, allows the use of not only hydrogen but also simple hydrocarbon fuels, such as methane, biogas, methanol, ethanol, propane and butane. This process is called “internal reforming” since all (chemical) conversion of the fuel into components that can be electrochemically oxidized (hydrogen and carbon monoxide) take place in the fuel cell itself and reduce the amount of fuel preprocessing. In the carbonate ion fuel cell, carbon dioxide is needed to form the ion (equivalent to the provision of water in the hydroxide ion fuel cell), which is a specific feature of this type of fuel cell. The internal reforming reaction of carbonate ion exchange fuel cells for methane is CH4 1H2 O-3H2 1 CO

(1.x)

The hydrogen molecules produced in this reaction take apart in electrochemical reaction with the carbonate ions diffusing from the cathode side as follows: 2 H2 1CO22 3 -H2 O1CO2 12e

(1.xi)

The free electron produced during this reaction travels to the cathode side via the external circuit, providing an electric current, and reduces a carbon dioxide molecule to form the carbonate ion as follows: 1 O2 1CO2 12e2 -H2 O1CO22 3 2

(1.xii)

8

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Using the CO molecule from Eq. (1.x) in place of hydrogen in Eq. (1.xi) results in direct oxidation to two CO2 molecules (plus electrons), showing that CO acts as a fuel for direct electrochemical oxidation. Fig. 1.6 depicts a carbonate ion exchange fuel cell. The carbon dioxide produced at the anode side can be recycled to the cathode side to supply the CO2 required there for forming the carbonate ions. Without a continuous supply of carbon dioxide, the carbonate ion fuel cell cannot convert hydrogen fuel, for instance. The molten carbonate fuel cell is the example for a carbonate ion-based fuel cell.

1.4 Thermodynamics of fuel cells The “fuel” input to fuel cells is the chemical energy contained in the H2 and O2 feeds, while the electrical energy is the desired output. Among the various chemical energy terms, enthalpy, Helmholtz function, and Gibbs free energy are key to fuel cell efficiency calculations. The Gibbs free energy, also known as free enthalpy, is a thermodynamic potential that determines the maximum reversible work of a fuel cell, the isothermal, isobaric with no work exerted by boundary or pressure forces on the thermodynamic system. The Gibbs free energy of

Figure 1.6 Schematic of a carbonate ion exchange fuel cell.

Introduction to fuel cells

9

formation is not constant and depends on temperature and state of the substances. The Gibbs free energy change is related to enthalpy and entropy change by ΔG 5 ΔH 2 T ΔS

(1.1)

where G, H, and S are the Gibbs free energy, enthalpy, and entropy, respectively. In all types of fuel cells running on hydrogen fuel, the chemical reaction between hydrogen and oxygen takes place and water is produced. The energy released during the chemical reaction is determined by the term Gibbs free energy of formation, Gf, instead of the Gibbs free energy. The energy released, Gf, is equal to the difference between the Gibbs free energy of the product(s) and the Gibbs free energy of the reactants as follows: ΔGf 5 Gf ;prod 2 Gf ;react

(1.2)

where subscripts “f,” “prod,” and “react” refer to formation, products, and reactants, respectively. It is more convenient to use the molar thermodynamic properties in chemical reactions for a better comparison. The molar quantities are indicated by a dash line over the specific properties such as the molar Gibbs free energy for water formation, ðgf ÞH2 O . In addition, a standard reference state, which commonly is 25°C (77°F) and 1 atm (STP, standard temperature and pressure), is used to calculate the changes in the energy of a system during the reaction process at well-defined conditions. The properties at the standard reference state are represented by a superscript (°) such as, for example, in the Gibbs free energy at the standard reference state (STP), G°. For a simple hydrogen/oxygen fuel cell the overall reaction is written as 1 H2 1 O2 -H2 O 2

(1.xiii)

According to Eq. (1.xiii) one mole of H2 and half a mole of O2 as reactant produce water that is called product. The change in molar Gibbs free energy of formation, Δgf , is expressed as follows: 1 Δgf 5 ðgf ÞH2 O 2 ðgf ÞH2 2 ðgf ÞO2 2

(1.3)

10

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

where ðgf ÞH2 O ; ðgf ÞH2 ; 12 ðgf ÞO2 are the molar Gibb’s free energy of formation of water, hydrogen, and oxygen, respectively. Combining Eqs. (1.1) and (1.3) gives the molar Gibbs free energy of formation as Δgf 5 Δhf 2 T Δs

(1.4)

Similarly, the value of the molar enthalpy change, Δhf , and molar entropy change,Δs, are equal to the molar enthalpy and molar entropy of products minus the molar enthalpy and molar entropy of reactants as follows, respectively: Δhf 5 hf ;product 2 hf ;reactant

(1.5)

Δs 5 sproduct 2 sreactant

(1.6)

and

Combing the chemical reaction of water formation, Eq. (1.3), with Eqs. (1.5) and (1.6), the molar enthalpy change and the molar entropy change are rewritten as follows, respectively: 1 Δhf 5 ðhf ÞH2 O 2 ðhf ÞH2 2 ðhf ÞO2 2

(1.7)

1 Δs 5 ðsÞH2 O 2 ðsÞH2 2 ðsÞO2 2

(1.8)

In high-temperature operating fuel cells, such as SOFCs, all gas mixtures behave as ideal gas mixtures. This means that the thermodynamic properties are a function of temperature only, not of pressure. Therefore, the values of hf and s that are a function of molar heat capacity, c p , vary with temperature only as follows, respectively [1]: hf ðT Þ 5 h298:15 1

ðT c p dT

(1.9)

1 c p dT 298:15 T

(1.10)

298:15

sðT Þ 5 s298:15 1

ðT

The molar heat capacity for high-temperature fuel cells, such as SOFC, is generally determined empirically and is usually given in texts and literature. An example of such an equation is presented in the study of Todd and Young [2]. The empirical equations for the molar heat capacity of hydrogen, oxygen, and water used in SOFCs over the absolute temperature range of 273
1473K and ambient pressure are given there as, respectively,

11

Introduction to fuel cells

0

1 0 12 0 13 T T T A 2 150:55@ A 1 199:29@ A c p;H2 ðkJ=kmol:KÞ 5 21:157 1 56:036@ 1000 1000 1000 0 14 0 15 0 16 T T T A 1 46:903@ A 2 6:4725@ A 2 136:15@ 1000 1000 1000

(1.11) 0

1 0 12 0 13 T T T A 1 203:68@ A 2 300:37@ A c p;O2 ðkJ=kmol:KÞ 5 34:850 2 57:975@ 1000 1000 1000 0 14 0 15 0 16 T T T A 2 91:821@ A 1 14:776@ A 1 231:72@ 1000 1000 1000

(1.12) 0

1 0 12 0 13 T T T A 1 146:01@ A 2 217:08@ A c p;H2 O ðkJ=kmol:KÞ 5 37:373 2 41:205@ 1000 1000 1000 0 14 0 15 0 16 T T T A 2 79:409@ A 1 14:015@ A 1 181:54@ 1000 1000 1000

(1.13)

By combining and integrating Eqs. (1.9) through (1.13), the value of hf and s of oxygen, O2, hydrogen, H2, and water, H2O, at specified temperatures are determined. Table 1.1 tabulates the values of enthalpy and absolute entropy [3]. The overall chemical reaction of carbon monoxide oxidation in hightemperature fuel cells is as follows: 1 CO 1 O2 -CO2 2

(1.xiv)

Table 1.1 Enthalpy of formation and absolute entropy at 25°C and 1 atm for hydrogen fuel cell [3]. Substance

Formula

h298:15 (kJ/kmol)

s298:15 (kJ/kmol  K)

Hydrogen Oxygen Water vapor

H2 (g) O2 (g) H2O (g)

0 0 2 241,820

130.68 205.04 188.83

12

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Similarly, the molar Gibbs free energy of the carbon monoxide reaction is calculated by Eq. (1.3). The change of the molar enthalpy and entropy is determined as follows, respectively: 1 Δhf 5 ðhf ÞCO2 2 ðhf ÞCO 2 ðhf ÞO2 2

(1.14)

1 Δs 5 ðsÞCO2 2 ðsÞCO 2 ðsÞO2 2

(1.15)

Again, Eqs. (1.9), (1.10), and (1.12) along with data from Ref. [2] are used to estimate the molar enthalpy and entropy of each chemical species as follows, respectively: ! !2 !3 T T T c p;CO ðkJ=kmol:KÞ 5 30:429 2 8:1781 1 41:974 1 5:2062 1000 1000 1000 !4 !5 !6 T T T 2 66:346 1 37:756 2 7:6538 1000 1000 1000

(1.16) ! !2 !3 T T T c p;CO2 ðkJ=kmol:KÞ 5 4:3669 1 204:60 1 657:88 2 471:33 1000 1000 1000 !4 !5 !6 T T T 2 519:9 1 214:58 2 35:992 1000 1000 1000

(1.17)

Table 1.2 lists the values of enthalpy and absolute entropy for carbon monoxide oxidation [3]. Following the calculation of Δgf , the value for the overall reaction of the fuel cell electrical work can be determined. The maximum amount of

Table 1.2 Enthalpy and absolute entropy at 25°C and 1 atm (STP) for carbon monoxide as a fuel [3]. Substance

Formula

h298:15 (kJ/kmol)

s298:15 (kJ/kmol  K)

Carbon monoxide Oxygen Carbon dioxide

CO (g) O2 (g) CO2 (g)

2 110,530 0 2 398,520

197.65 205.04 213.8

Introduction to fuel cells

13

electrical work extracted from a fuel cell reaction is derived from Gibb’s thermodynamic relation as follows: G 5 H 2 TS 5 U 1 pV 2 TS

(1.18)

By differentiating Eq. (1.18) and using the first law of thermodynamics for closed systems, the total work transferred between the system and the surroundings is dW 5 TdS 2 dU 5 dG 1 TdS 1 SdT 2 pdV 2 Vdp

(1.19)

The total work term includes the boundary work (b) and electrical work (e): dW 5 dWb 1 dWe

(1.20)

As known from thermodynamics of reversible processes, the boundary work is written as “ 2 pdV.” Combining Eqs. (1.19) and (1.20) and considering isothermal (dT 5 0) and isobaric (dp 5 0) processes, the maximum electrical work for a fuel cell system is determined by dWe 5 2 dG

(1.21)

We 5 2 Δg

(1.22)

Integrating Eq. (1.21) gives where Δg is the molar Gibbs free energy of formation. Faraday’s law for standard-state condition states that [4] We 5 E0 I 5 E0 nF

(1.23)

where E0 is the standard-state reversible voltage, n is number of electrons passing the external electrical circuit, and F is Faraday’s constant. Plugging Eq. (1.23) into Eq. (1.22) gives E0 5

Δg0 nF

(1.24)

where Δg0 is the standard-state Gibbs free energy change for the overall reaction of the fuel cell. For example, the reversible voltage generated under the standard conditions for a hydrogen
oxygen fuel cell is calculated from E0 5

Δhf 2 T Δs ðhH2 O 2 hH2 2 12 hO2 Þ 2 T ðsH2 O 2 sH2 2 12sO2 Þ Δg0 5 5 nF nF nF (1.25)

14

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Using Table 1.1 data, Eq. (1.25) E0 becomes E0 52

ð2241820243950:6202 12 30Þ22983ð188:832130:682 12 3205:04Þ 51:23 V 2396485 (1.26)

The maximum voltage produced by a hydrogen
oxygen fuel cell is determined by subtracting the enthalpy of formation of water vapor (given in Table 1.1) from the enthalpy of vaporization of water at 298K (i.e., 43950.6 kJ/kmol). This is due to state of water which is liquid at STP condition. Therefore, the maximum voltage that is produced by the hydrogen
oxygen fuel cell at STP is 1.23 V. The equation for an arbitrary chemical reaction on a molar basis is as follows: 1A 1 bB"cC 1 dD

(1.27)

where A and B are reactants, C and D are products, and 1, b, c, and d represent the stoichiometric coefficient of species A, B, C, and D, respectively. The reversible cell voltage as a function of concentration of the reaction species is determined by the Nernst equation as follows: E 5 E0 2

 c d RT c c ln C1 Db nF cA cB

(1.28)

where ci denotes the concentration of ith species, R is the universal gas constant and is equal to 8.314 J/mol  K, F is Faraday’s constant and is equal to 96485 C/mol, and n is the number of electrons taking part in the electrochemical reaction. E0 is the reversible voltage under standard-state conditions and is obtained by Eq. (1.24). By combining Eq. (1.28) and the ideal gas equation (pv 5 RT), the reversible cell voltage in terms of partial pressure of each species is determined by E 5 E0 2

 c d RT p p ln C1 Db nF pA pB

(1.29)

Chapter 2, Classification of SOFCs, covers SOFC modeling based on the equations that are extracted from thermodynamic relations and discusses their behavior. Also, the efficiency of a SOFC will be covered in Chapter 2, Classification of SOFCs, using these thermodynamics relationships.

Introduction to fuel cells

15

References [1] R.T. Balmer, Modern Engineering Thermodynamics, first ed., Elsevier, 2011. [2] B. Todd, J.B. Young, Thermodynamic and transport properties of gases for use in solid oxide fuel cell modelling, J. Power Sources 110 (2002) 186
200. [3] Y.A. Çengel, M.A. Boles, Thermodynamics: An Engineering Approach, fifth ed., McGraw-Hill, 2006. [4] J. Larminie, A. Dicks, Fuel Cell Systems Explained, second ed., John Wiley & Sons, 2003.

CHAPTER 2

Classification of solid oxide fuel cells Contents 2.1 Historical summary 2.2 Geometrical types 2.2.1 Planar design 2.2.2 Tubular design 2.2.3 High-power density design 2.2.4 Delta design 2.2.5 Button design 2.3 Cell types in terms of its support 2.3.1 Electrolyte-supported solid oxide fuel cell 2.3.2 Cathode-supported solid oxide fuel cell 2.3.3 Anode-supported solid oxide fuel cell 2.4 Solid oxide fuel cell classification based on flow patterns 2.5 Cell types in terms of its chamber number 2.5.1 Dual-chamber solid oxide fuel cell 2.5.2 Single-chamber solid oxide fuel cell 2.5.3 No-chamber solid oxide fuel cell 2.6 Single and stack cell designs References

18 21 21 22 23 24 25 27 28 28 30 32 36 36 37 41 42 45

Solid oxide fuel cell (SOFC) development was first discovered by Nernst in 1899 when he introduced zirconia (ZrO2) as an oxygen ion conductor. The SOFC mechanism, solid and gas phases, is the simplest compared to other types of fuel cells like molten carbonate fuel cells. The SOFC, a fully solid-state device, converts the chemical energy of its fuel by the oxide ion conducting ceramic material of its electrolyte to electrical power. In SOFC, the negatively charged ion (O22) is transferred from the cathode through the electrolyte to the anode and produces water at the anode side.

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells DOI: https://doi.org/10.1016/B978-0-12-815753-4.00002-6

© 2020 Elsevier Inc. All rights reserved.

17

18

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

SOFC, in addition to other fuels, can use hydrogen and carbon monoxide. SOFC compared with other fuel cells such as phosphoric acid and molten carbonate fuel cell does not need the electrolyte management. Finally, the high-operating temperature of SOFC eliminates the need for the precious metal electrocatalysts. In following section, the most common classes of SOFC are described and presented. The main focus of this chapter is on the dual-chamber SOFC, also known as the conventional SOFCs, due to its considerable efficiency. The limitations of the conventional type of SOFC such as gas leakage and cracks caused by thermal stresses are discussed and ways to overcome these limitations are presented. The single-chamber SOFC, where the mixture of fuel and oxidant is directly fed to the fuel cell, as a novel alternative to the conventional type is introduced and the way it operates is discussed in detail. Another class of SOFCs that is described in this chapter is no-chamber SOFC. It is also known as direct-flame SOFC because the flame and fuel cell are geometrically and electrochemically coupled. Furthermore, other types of fuel cells based on their classifications such as geometry, flow patterns, and support types are presented and are discussed.

2.1 Historical summary As stated in Chapter 1, Introduction to fuel cells, both SOFC and molten carbonate fuel cells can use different fuels, such as hydrogen and methane, due to their high-temperature operation (about 1000°C). After the discovery of zirconia (ZrO2) as an oxygen ion conductor by Nernst in 1899, Swiss scientists Emil Baur and his colleague Hans Preis experimented the first SOFC in the late 1930s. Baur and his colleague used an electrolyte made from zirconium along with yttrium, cerium, lanthanum, and tungsten material. During their experiment, undesirable chemical reactions took place between electrolyte and carbon monoxide, which caused the electrolyte to corrode and the electrical conductivity to decrease. Davtyan, a Russia scientist, tried to take care of the problem with the cell electrical conductivity created during the Baur and Preis experiment. Davtyan added monazite sand to a mix of sodium carbonate, tungsten trioxide, and soda glass in order to increase the conductivity and mechanical strength of the electrolyte material. Again Davtyan experienced unwanted chemical reactions which shortened the life ratings of his cell.

Classification of solid oxide fuel cells

19

By the late 1950s research on SOFC picked up a very fast momentum. Scientists from different countries and institutes, such as the Central Technical Institute of Netherlands and Consolidation Coal Company and General Electric of United States, started working on SOFC. After several years of research on SOFC, in 1959 scientists noted problems such as relatively high internal electrical resistance, melting, and short-circuiting due to semi-conductivity with the SOFC solid electrolytes. These problems did not distract the researchers’ focus on SOFC enhancement due to its high-operating temperature criteria. Therefore, research on resolving the existing problems on SOFC continued. For instance, scientist at Westinghouse experimented a SOFC that contained zirconium oxide and calcium oxide in its electrolyte material. In 1961 they revealed their new patented idea related to solid-electrolyte based fuel cell for measuring the oxygen concentration of a gas phase with a concentration cell [1,2]. Their findings helped the Westinghouse scientist to develop and successfully test the first tubular “bell-and-spigot” SOFC stack. The development by the Westinghouse scientists eventually became the foundation of today’s cathode-supported, tubular seal-less SOFCs. In parallel effort, other scientists advanced their researches on the material used in SOFC electrode specially that of cathode material. The new material tested by scientists initially was platinum, then transitioned to doped In2O3 [3] and finally settled on what is known today as doped LaMnO3. This development was achieved because scientists could effectively activate the oxygen-reduction process using the rare-earth, transition-metal perovskite oxides as cathode material due to its unique electrical and catalytic properties. Further researches by scientists produced a new SOFC cathode material known as doped LaMnO3. This material has excellent thermal-expansion property. In addition, Meadowcroft found another important material for SOFC interconnect, called doped LaCrO3 perovskite [4]. Furthermore, Spacil revealed his patented composite anode material made of Ni metal with a ZrO2-based electrolyte. To date the Ni metal with a ZrO2-based electrolyte is the standard material used for anode of the SOFC [5]. In 1970s the electrochemical vapor deposition invention by Arnold Isenberg of Westinghouse Corporation brought about a revolution in technical development of SOFCs. Using the electrochemical deposition technique, Isenberg deposited a perfectly dense ZrO2 electrolyte thin film on the substrate of a porous, tubular substrate at relatively low temperatures. Originated from Isenberg invention, the Westinghouse Corporation

20

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

successfully manufactured and tested a series of modern SOFC generators in the ranges of 5 250 kW. Inspired by these new and modern SOFCs development, various SOFC stack designs such as tubular and planar type were designed and built. In addition, alternative materials for the SOFC cathode, anode, and interconnect were introduced and used in related substrates. Souza and his colleague of Berkeley National Laboratory advanced the anode-supported planar SOFCs in 1997 [6]. They essentially demonstrated that an electrolyte on a porous anode substrate can be co-fired at high temperatures into a dense thin film without invoking chemical reactions. Then the cathode was also sintered at much lower temperatures to minimize chemical reactions. As a result, the single-cell performance improved significantly which in turn allowed the anode-supported SOFC to operate at lower temperatures where commercially available oxidationresistant alloys such as thermal expansion compatible ferritic steels can be utilized as interconnect materials for SOFC stacks. A majority of modern SOFCs adopt the anode-supported planar geometry to reduce cost and increase performance. It does not have anything to do with reliability and stability which are the leading issues for commercialization at the present time. Since the discovery by Nernst over 100 years ago, the ZrO2-based materials is still the main material for SOFC electrolytes. Goodenough and his colleagues [7] study revealed that perovskites had high oxide-ion conduction in other structures than the classical fluorite structure, giving hopes for finding a new family of oxide-ion conductors in other crystal structures. This prediction was favorably vindicated by the noteworthy discovery of the high oxide-ion conductivity perovskite Sr- and Mg-doped LaGaO3 (LSGM) by Ishihara et al. [8] in 1994, immediately confirmed by Feng and Goodenough [9] in the same year, followed by a systematic characterization of the system by Huang et al. [10,11]. The high oxide-ion conductivity and the crystallographic compatibility make the perovskite LSGM an attractive cathode material for lowtemperature SOFCs. Mitsubishi Materials Incorporation tested a SOFC with LSGM electrolyte at operating temperature of 800°C and demonstrated an excellent stack performance [11]. The United States Department of Energy 220 kW SOFC, that was running with natural gas, showed 60% efficiency under normal operation condition. In addition a cogeneration plant with a 140 kW peak power SOFC, supplied by Siemens Westinghouse, is presently operating in the

Classification of solid oxide fuel cells

21

Netherlands. This system has operated for over 16,600 h, becoming the longest running fuel cell in the world [12]. Furthermore, USDOE and its partner Westinghouse Corporation also worked on building and operating a cogeneration plant with 1 MW SOFC [13]. In summary, the major driver for sustaining the development of highoperating temperature SOFC technology is its intrinsically high electrical efficiency compared with a conventional heat engine. In addition, SOFC showed the ability to withstand more than 100 thermal cycles and voltage degradation of less than 0.1% per thousand hours.

2.2 Geometrical types The most common SOFCs classification based on cell geometries according to manufactures are (1) planar-type SOFC, (2) tubular-type SOFC, and (3) button-type SOFC. However, more general classification based on geometrical design is involved and the detail of each types are provided in the following sections.

2.2.1 Planar design Planar, also known as flat plate, SOFCs are more popular than the tubular type of SOFCs because they are easy to fabricate, operate at a lower temperature, and offer a higher-power density [14]. A typical unit cell in a planar SOFC is made of a positive electrode electrolyte negative electrode, also called PEN, assembly, a porous nickel mesh, two end interconnect plates and gas seals, as shown in Fig. 2.1. As can be seen, in a Upper interconnect Anode Electrolyte Cathode

Fuel Air

Lower interconnect

Figure 2.1 Schematic of a planar-type SOFC.

22

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

planar SOFC-type configuration, the electrolyte layer is sandwiched between anode and cathode layers. In some collector plates, the air and fuel is supplied through the channels that are embedded in the cell flat plate. This design reduces the Ohmic resistance and gives higher-power densities compared to tubular type of SOFC. Another advantage of the planar-type SOFC is its simple design and low-cost fabrication method such as screen printing and tape casting that reduces the manufacturing costs. Planar substrate is normally fabricated by tape casting and other components are doped on the surface by screen printing, and/or laminating process. One of the major disadvantages of planar design is the reactants leakage through the cell edges, requiring gas-tight sealing. The sealing material must not only seal the edges of the cell and avoid fuel leakage and air mixing but also be mechanically compatible (i.e., must match the coefficient of thermal expansions with adjacent components) and chemically stable (i.e., long-term stability under the RedOx environment). Glass ceramics and glass are materials that are used for preventing the leakage in planar-type SOFCs [15].

2.2.2 Tubular design The tubular SOFC was first introduced in late 1970s by the United States Westinghouse Electric Corporation, currently called Siemens Westinghouse Power Corporation. Tubular shape is the most advanced type of SOFC. A typical view of the tubular SOFC is shown in Fig. 2.2. Interconnection

Electrolyte

Fuel flow

Cathode

Anode Air flow

Figure 2.2 Schematic of a tubular-type SOFC.

Classification of solid oxide fuel cells

23

Siemens Westinghouse Power Corporation demonstrated a 200 kW tubular-type system with ceramic tubes manufactured by extrusion, sintering, and coating. The way they manufacture the tubular part of the SOFC is to process and form the ceramic substrate by extrusion into a tube. Then the remaining components are deposited onto the tube surface by masking processes. One advantage of the tubular design is of its high mechanical stability. On the other hand, it has low-power density compared with planar design because of high internal resistance and high electrical resistance due to longer current paths. For instance, a planar SOFC is capable of producing 2 W/cm2 at operating temperature of 1273K, while a tubular one at the same operating condition can produce only about 0.25 0.3 W/cm2 [16].

2.2.3 High-power density design In the mid-1990s Westinghouse Electric Corporation introduced a new type of SOFC called high-power density (HPD) fuel cell. The new design was able to overcome the tubular high Ohmic resistance resulting from the long current pathway along the circumference of the circular cross section [17]. The important feature of this new cell is the new current pathway design. The pathway was shortened by using a rectangular shape cross section instead of circular cross section. In addition, multiple ribs were used to connect the two flattened surfaces, while the air cavity channels design did not change. In the literature, the cell designation was often seen attached with a figure denoting the number of air channels. For example, HPD8 denotes for a HPD type of SOFC having eight air channels. Fig. 2.3 depicts the schematic of a HPD design having five air channels. Over the years, engineers investigated the effect of HPD geometrical cell parameters on its electrical performance and thermal stress distribution. Optimization of a HPD cell geometry involves with all aspects of electrochemistry and mechanical process. The parameters that affect the electrical performance and mechanical strength of the HPD fuel cell are cathode substrate design such as cell width, cell wall thickness, and number of ribs and their heights. For example, a wider and thinner HPD cell increases the surface area and reduces the current path length and therefore produces more power at higher-power densities. Drawbacks to this design are the mechanical strength and durability, which is compromise, and smaller air channels that could elevate the air delivery pumping power, which reduces the net efficiency of the system. In the late 1990s

24

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Fu e

l fl

ow

Interconnect/contact

Anode

Electrolyte

Air flow

Cathode

Figure 2.3 A schematic illustration of HPD design.

to early 2000s, engineers used the optimizing techniques and designed new cells called HPD 5 and 10. The new HPD5 design featured larger air channels and longer current path length [17]. This cell type has the same functional layers as a tubular SOFC and the oxidant can be delivered in three different ways. Specifically, (1) by means of anode feed tubes as is custom for the standard tubular SOFCs, (2) by using some of the channels for the air inlet and the others for exhausting the vitiated air, and (3) by passing the air once through with a cell that is open at both ends.

2.2.4 Delta design Delta cell is an alternative version to the HPD cell design that was developed by the Siemens/Westinghouse during the 2000s. Since the Delta cell active surface area is wavy (i.e., triangular) and larger, it increases the power output without compromising the specific power density [17]. In addition, the triangular geometry of the Delta cell automatically provides for both air and fuel to flow through. The adoption of Delta8 (eight air channels) design represented a significant increase in cell surface area. However, since it is shown that the stress concentration points are usually located at the valley joints between two adjacent triangles, the increase in surface area and therefore in total power output comes at the expense of compromising the cell mechanical strength. Therefore, the Delta cells are generally considered less robust in comparison to HPD cells. Fig. 2.4 depicts the schematic of a Delta8 design.

Classification of solid oxide fuel cells

25

Fu e

l fl

ow

Interconnect/contact

Anode

Electrolyte

Air flow

Cathode

Figure 2.4 A schematic illustration of Delta8 design.

2.2.5 Button design The button design of solid oxide fuels can appear in both planar and tubular configuration. The button configuration in the planar and tubular type of SOFC are shown in Fig. 2.5A and B, respectively. The anode and cathode channel of the button-type fuel cell are separated by what is called positive-electrode/electrolyte/negative-electrode (PEN). The PEN is sealed in a way that prevents the gas leakages. Researches have tried smaller tubes with smaller diameters in many of their single-chamber solid oxide experiments. Smaller tubes are used to transport the fuel, hydrogen or hydrocarbons, and oxidant, usually air, to the anode and cathode, respectively (describing in future section by defining a vertical PEN in the flow channel that extends to both the top and bottom walls with the addition of several inert objects (walls) to represent the tubes for gas delivery). The button-cell design is mostly used in laboratory experiments that is applicable in performing SOFCs tests in order to achieve cell performance enhancement in labs. As an applicable case, for planar SOFCs testing, achieving good cell sealing and proper contact between electrodes and current collecting components is crucial for the right results with as little external interference as possible. In this case, an alumina test kit may be taken into consideration when it comes to assessing the commercial feasibility due to its reasonable thermal expansion coefficients, which will be first chosen if one is going to evaluate the long-term performance. However, to develop new materials with various purposes, it is more

26

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Air Anode

Electrolyte

Cathode

Cathode

Anode Electrolyte Fuel (A)

(B)

Figure 2.5 Schematic illustration of (A) planar fuel cell in the button-cell configuration and (B) tubular fuel cell in the button-cell configuration.

practical to test a button cell with regard to fundamental research or to test a commercial single cell as a real short stack. Accurate evaluation of the performance of a button cell is needed in a lab which focuses on fundamental research. There are mainly two factors, including cell sealing and contact between each component, which determine the testing results. The input and output gas flow can be simultaneously monitored to judge the air-tightness of the whole testing kit. A bubbler is sometimes applied to roughly check the output gas flow. Special glass and ceramic sealant paste are often recommended for SOFCs to metals and ceramics, and extra compaction force is usually needed to achieve a better sealing effect. A proper kind of paste plus the right force will make an open circuit voltage (OCV) close to 1.2 V if the electrolyte is dense enough. A simple kit for button-cell testing can be designed and manufactured as shown in Fig. 2.6. Two inner thin gas tubes with four holes are usually selected not only to input air and fuel but also to accommodate the platinum cables for voltage and current collection. If the temperature distribution along the furnace has a gradient, it can be monitored using a thermocouple to detect the temperature point close to the cell position. It should be noticed that the low-level content of sealing glass (SiO2) tends to contaminate the cell electrodes, so it is better to fabricate the cell with larger electrolyte areas to avoid the cell directly touching the electrode.

Classification of solid oxide fuel cells

27

Air in Pt wire

Inner tube Air out

Furnace

Out tube Sealing Cell Pt mesh Fuel out

Stainless three-way support

Fuel out

Pt wire

Fuel in

Figure 2.6 A schematic illustration of a button-cell design used for a test.

2.3 Cell types in terms of its support Based on component thickness, the SOFC is classified into two broad categories: the self-supporting configuration and the external-supporting configuration [18]. In the self-supporting configuration one of the cell components, mostly the thickest layer, acts as the cell structural support. Thus, the self-supporting cell can be designed as electrolyte-supported, anode-supported, and cathode-supported or metal-supported fuel cells. In an electrode-supported SOFC either the anode, anode-supported, or the cathode, cathode-supported, is thick enough (normally between 0.3 and 1.5 mm) to serve as the supporting substrate for the cell. An advantage for self-supporting cell is its applicability for SOFCs operating on reduced temperature range. Highly active electrolyte materials with polarizations reduce the operating temperature to an intermediate range, 600°C 800° C, while it increases both the Ohmic and activation polarizations of the electrodes. This range of operating temperature can overcome the problem associated with high-operating temperature of a typical SOFC. Typical SOFC, also referred to as intermediate-temperature SOFCs (ITSOFCs), operate between 823K and 1073K. Intermediate SOFCs are electrode-supported type [19]. Other losses such as Ohmic loss due to ion

28

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

transport from the electrolyte are significantly reduced by electrodesupported design because this cell type have very thin electrolyte, as thin as 10 µm, which leads to lower Ohmic loss [19]. Much efforts are devoted by scientists to develop new materials and configurations to improve the performance of SOFCs by reducing the operating temperatures [20]. At low-operating temperature the anode-supported SOFCs are preferred over the cathode-supported one because the cathode-supported design is susceptible to significant activation and concentration over-potentials. On the other hand, in the external-supporting configuration, the single cells are supported either by the interconnect or by a porous substrate. One of the most important design criterions for a functioning fuel cell is the separation of anode and cathode by a gas-tight electrolyte. Pinholes or cracks in the electrolyte can cause the hydrogen to leak into the cathode compartment where it reacts directly with oxygen. This phenomenon, which occurs in anode-supported cells with very thin electrolyte films, decreases the OCV and might even cause the fuel cell to stop operating. In the following a description of different SOFC self-supported configurations is provided.

2.3.1 Electrolyte-supported solid oxide fuel cell Most early SOFC prototypes used the electrolyte-supported PEN design. In this scheme, the electrolyte is the thickest part of the electrolytesupported PEN design, less than 100 µm and acts as the structural support for the entire PEN. To make the electrolyte, first the material is prepared and then undergoes the heat/fire test to achieve its strength. Subsequently a thin, porous anode and cathode electrode layers are deposited on either side of the electrolyte by spray coating, dip coating, or tape casting. Finally, the PEN is heated/fired to possess good mechanical properties. Electrolyte-supported SOFCs are typically mechanically strong and resistant to delamination and thermal shocking. Fig. 2.7A and B depict the schematic of the planar and tubular cathode-supported SOFC configuration, respectively.

2.3.2 Cathode-supported solid oxide fuel cell The thickest part of the cathode-supported SOFC is cathode layer which is less than 1 mm and acts as support for the entire PEN structure. To manufacture porous cathode structure of this thickness, start by mixing the cathode electrode powder, typically LSM, with binders and pore

Classification of solid oxide fuel cells

29

Interconnection

Electrolyte

Fuel flow

Cathode

Anode Air flow

(A) Upper interconnect Anode Electrolyte Cathode

Fuel Air

Lower interconnect

(B) Figure 2.7 A schematic illustration of an anode-supported SOFC in (A) planar configuration and (B) tubular configuration.

former, typically carbon black or starch. Then the material is extruded, die pressed, or tape casted to form the desired cathode shape. Subsequently, a 10 30 µm finely textured and mixed interfacial layer of LSM 1 YSZ is applied to one side of the cathode to create a large number of triple-phase boundary sites by intimately mixing the ionconducting (YSZ) and electrically conducting (LSM) phases. Another characteristic of this layer is its porosity which is finer than the cathode thick structure part of the cathode. Therefore, it is easier to do the next

30

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

step in producing the material which is the deposition of a dense layer on the electrolyte. Next step is to mix the electrolyte powder (typically YSZ) with dispersants and a solvent (typically an alcohol) to come up with an electrolyte slurry. Then, a thin and dense layer of the slurry (,20 µm) is applied on top of the cathode by spray coating, dip coating, or screen printing and immediately the cathode 1 electrolyte structure is heated/ fired in order to possess good mechanical properties. Extreme care is usually needed during the deposition and heating of the electrolyte layer to ensure a healthy and defect-free structure. Finally, the same procedure is used to create the anode slurry by typically applying a thin layer of YSZ 1 NiO powders, less than 200 µm, on top of the electrolyte by spray coating, dip coating, or screen printing. Again following after the anode application, the PEN is heated/fired to ensure healthy structure. It is worth noting that the cathode-supported PENs mechanical properties, such as cracking or delamination, is not as good as the mechanical properties of the electrolyte-supported PENs. The thin structure of the electrolyte-supported PEN dramatically reduces the cell Ohmic resistance, while the cathode thickness considerably increases the oxygen mass transport resistance which in turn increases the oxygen mass transport losses. In general, reduction of Ohmic resistance outweighs the increase of the oxygen mass transport resistance. Therefore, the cathode-supported PENs outperform the electrolyte-supported PENs. Fig. 2.8A and B depict the schematic of a planar and tubular cathode-supported SOFC configuration, respectively.

2.3.3 Anode-supported solid oxide fuel cell The thickest part of the anode-supported SOFC is anode layer which is less than 1 mm and acts as the entire PEN structural support. The processing procedure is basically identical to the cathode-supported PEN approach. The only exception is that it starts with the creation of the thick, porous anode structure which is formed by a mixture of NiO and YSZ powders. Then a thin and fine-textured with reduced porosity is deposited on the anode layer and after that on electrolyte and cathode layers, respectively. Again, the anode-supported PENs mechanical properties, such as cracking or delamination, are not as good as the mechanical properties of the electrolyte-supported PENs. However, even though the fuel mass transport resistances of the anode-supported PENs increase considerably, it does not concern the researchers as much as the problem of

Classification of solid oxide fuel cells

31

Interconnection

Electrolyte

Fuel flow

Cathode

Anode Air flow

(A)

Upper interconnect Anode Electrolyte Cathode

Fuel Air

Lower interconnect (B)

Figure 2.8 A schematic illustration of an anode-supported SOFC in (A) planar configuration and (B) tubular configuration.

the cathode-supported oxygen mass transport problems should. It is for two reasons: (1) The anode is typically supplied with 100% fuel, whereas oxygen at the cathode is already diluted to 21%, since air contains only 21% oxygen, and (2) the percent of fuel conversion is higher before it reaches the electrochemically active interface. This is because the resident time of the hydrocarbon fuel that is directly supplied to the thick anode structure increases. The increase in anode thickness improves the hydrocarbon-fueled fuel cell performance. Therefore, the anode-supported PENs show the best performance among the three mentioned PEN designs and are being

32

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

intensively developed by both industrial and academic researchers. The similar finding of this report comes up with the recent results in Kamvar et al. [21] work in the case of single-chamber SOFC. In this research, the oxygen mass transport problems in cathode-supported cell design in some extent is due to amount of oxygen content at cathode side of the cell. The mentioned problem can stop the operation of the SOFC specially in single chamber scheme. In an anode-supported SOFC, the highest potential loss is due to Ohmic over-potential. The other losses, that is the cathode and electrolyte over-potentials, are not negligible. Therefore, as opposed to selfsupported SOFC the anode-supported fuel cell is the potential candidate for the moderately operating fuel cell design. Fig. 2.9A and B shows the schematic of the planar and tubular anode-supported SOFC configuration, respectively. Table 2.1 shows the thickness range of components of different types of PEN-supported design.

2.4 Solid oxide fuel cell classification based on flow patterns The influence of different polarizations, potentials, varies depending on the cell structure, design, and operating conditions. The different potentials vary along the flow direction and are affected by the local temperature as well as the local hydrogen and oxygen concentrations. For instance, the cathode activation polarization is the dominant potential in a co-flow SOFC with intermediate temperature near the fuel and air inlets, and in the anode near the outlets. The variation of the cathodic activation over-potential in the counter-flow SOFC configuration is smaller compared to that of the co-flow configuration. The cathode activation polarization is dominant in both co-flow and counter-flow SOFC configurations, especially at low-current densities, while the anode activation polarization is dominant at the higher-current densities. The dominance of the anode activation at high-current densities is due to hydrogen depletion of high-fuel utilization. The influence of the ionic polarization depends mostly on the ionic transfer conductivity within the electrolyte and the electrolyte thickness. The decrease in the electrolyte thickness decreases the Ohmic polarization. When the fuel or oxidant concentration reaches zero at the three-phase boundary, the concentration overpotentials increase while the voltage reaches zero.

Classification of solid oxide fuel cells

33

Interconnection

Electrolyte Fuel flow

Cathode

Anode Air flow

(A) Upper interconnect Anode Electrolyte Cathode

Fuel Air

Lower interconnect

(B) Figure 2.9 A schematic illustration of an anode-supported SOFC in (A) planar configuration and (B) tubular configuration.

Table 2.1 The thickness range of components of different types of PEN-supported design. Support types

Anode thickness range

Electrolyte thickness range

Cathode thickness range

Electrolyte-supported Cathode-supported Anode-supported

100 200 µm 100 200 µm 1000 2000 µm

200 500 µm 10 20 µm 10 20 µm

100 200 µm 1000 2000 µm 100 200 µm

34

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Fuel and air gas channels are formed on interconnects to distribute the reactants across the cathode and anode electrodes, respectively. Different flow field designs are possible depending on the relative positioning of the fuel and air channels, which can greatly affect distribution of the reactants’ concentration, temperature, and reaction rate. The interconnector acts as an important component of the SOFC. The design of interconnector is of significant importance to improve the performance of SOFC. There are two factors to consider when designing an interconnector: one is benefited to the gas transfer and reduce the concentration polarization. The other is to make the current path as short as possible and reduce the ohmic polarization before the current is collected. Air and fuel flow are directed through gas channels in the interconnect. The flow field configuration greatly determines the temperature and reaction rate distribution within the cell. General possibilities are crossflow, counter-flow, and co-flow configurations. In a rectangular crossflow configuration, fuel and air inlets and outlets are separated from each other, which allows for a simple manifold system. Therefore, it is easy to apply cross-flow in technical systems. Experimental cells are often circular with center gas inlets and radial co-flow configuration. However, counter-flow gives the flattest profiles for temperature and current. Mixtures between the flow patterns are, for example, Z-flow, serpentine flow for rectangular, and spiral-flow for circular configurations. Fig. 2.10 shows the most common flow configurations of a planar SOFC. Modern designs feature manifolding integrated into the cell plates. As stated, planar cells offer principally a high-volumetric power density and good electric performance through short current paths. Severe problems are, however, sealing of the cells and mechanical loads due to high thermal gradients. Many scholars have conducted a lot of researches to design the interconnector. Jiang et al. studied the relationship between the cathode/interconnector contact area and the cell performance. The result showed that the cell resistance dramatically reduced with the increase of the cathode/ interconnector contact area [22]. Lin et al. derived a formula to evaluate the influence of rib on the concentration loss [23]. Liu et al. established a model of anode-supported SOFC and found that the rib width had negligible influence on the transport of the anode side gas. At the cathode side, an anaerobic zone appeared [24]. Based on this finding, when designing the anode-supported SOFC interconnector, the anode rib size (relative to the cathode rib size) was increased in order to reduce the ohmic loss due to anode/interconnector contact resistance. Chen et al. proposed a

Classification of solid oxide fuel cells

35

Air counter-flow Air

Air Air co-flow Fuel

Fuel

Cross-flow

Fuel

Co/counter-flow Air

Air

Z-flow Air

Fuel Serpentine flow

Fuel

Fuel

Radial flow

Spiral-flow

Figure 2.10 Typical flow field configurations of a planar SOFC.

double-layer interconnector [25]. This design not only increased the flow rate of the gas in the airway but also enhanced the disturbance of the gas in the airway. Moreover, it improved the mass transfer in the porous electrode and effectively reduced the concentration loss. Luca et al. conducted experimental and theoretical analysis of the ray flow of a circular plate SOFC [26]. Shi et al. investigated a two-electrode-supported SOFC with a micro airway and plane interconnector [27]. It was found that this new design not only effectively enhanced the fuel gas and oxidizing gas but also made the structure of the SOFC stack more compact. Up to now, the straight channel interconnector is the most widely used configuration, which is defined as the conventional interconnector. Recently, Wei Kong et. al. [28] claimed that these conventional interconnectors are concentrated in the middle of the electrode, which are harmful for mass transfer and charge transfer. To solve this issue, they proposed a novel X-type interconnector, as depicted in Fig. 2.11. In this novel scheme, the whole interconnector is split up into many small X-type pieces and located staggered, which can promote the gas diffusion in the electrode and reduce the current path.

36

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

(A)

(B)

Figure 2.11 (A)A novel interconnector design of SOFC proposed by Wei Kong et al. [28]. (B) Exploded view of the cell.

2.5 Cell types in terms of its chamber number SOFCs work at high temperatures between 500°C and 950°C. These high-operating temperatures have both advantages and disadvantages. On one hand, they enable direct internal reforming at the anode as well as the fuel flexibility, on the other hand they can lead to crack formation within SOFC components due to thermal stresses. Also, the highoperating temperature gives rise to increased materials degradation. Several attempts have been done to pass these challenges related to SOFC commercialization. One of these attempts is to decrease the number of chamber numbers in order to liberate sealing and crack-formation issues. In this section, different SOFC configurations based on the number of its chamber are presented.

2.5.1 Dual-chamber solid oxide fuel cell In dual-chamber SOFCs (DC-SOFCs), which is also referred to as conventional SOFCs, fuel and oxidant have to be supplied separately to the respective electrode and any mixing of the two reactant gases has to be avoided. This avoiding of intermixing of fuel and oxidant guarantees the system safety and keeps the system apart from explosion risk. Also, it prevents diffusion of unwanted species into the relevant electrodes (e.g., it prevents to diffuse hydrogen into the cathode electrode where the presence of oxygen is necessary). Thus, this separate gas supply enhances cell performance. This is why the conventional SOFC is a more common design in SOFC commercialization.

Classification of solid oxide fuel cells

37

On the other hand, the necessity of separate gas supply imposes complex gas management and gas manifolding, thus not only complicating stack assembly but also downsizing SOFC systems. Moreover, gas-tight, hightemperature, and mechanically resistant sealing is inevitable in order to separate the cell into leakproof anode and cathode compartments. Sealing is generally achieved using glass or ceramic materials. The advantage of glass sealing is its rigidity and high gas tightness. However, degradation over time as well as thermal stresses during cycling often cause cracks. Sealing by mechanical compression with mica increases thermomechanical strength but results in higherleakage rates. In addition to the gas-tight sealing, the electrolyte has to be fully dense to avoid any gas cross-over between anode and cathode compartments.

2.5.2 Single-chamber solid oxide fuel cell In order to avoid challenges reported in the previous section, SOFCs with only one gas chamber, the so-called single-chamber SOFCs (SC-SOFCs), are being developed. SC-SOFCs are operated in mixtures of fuel and oxidant, thus completely eliminating any needs for gas sealing and any crack growth would not terminate their performance [21]. SC-SOFCs can be defined as fuel cells with only one gas compartment (thus single-chamber) operating in a non-equilibrium gas mixture of fuel and oxygen. This simplified design is titled as SC-SOFC and was made by Hibino and Iwahara for the first time in 1993 [29]. But “one-chamber,” “single-compartment,” “mixed-gas,” “mixed-fuels,” “mixed-reactant,” and “separator-free” fuel cells as well as “SOFCs with reaction-selective electrodes” can also be found in the literature. SC-SOFC is a novel type of SOFC in which the anode and cathode are both exposed to the same premixed fuel/air stream, and selective electrocatalysts are used to preferentially oxidize the fuel at the anode and reduce oxygen at the cathode. It operates in a mixture of hydrocarbon fuel and oxygen in which the amount of oxygen is less than that required for complete combustion of the fuel. This is called the fuel-rich condition and is required to produce hydrogen and CO (i.e., syngas) without producing significant amounts of CO2 and H2O. At the anode, selective catalysts result in in situ catalytic reactions (e.g., partial oxidation and reforming) of the fuel to produce the syngas, which are then electrochemically oxidized by reaction with oxygen ions at the anode electrolyte interface. At the cathode, gaseous oxygen is reduced to replenish the oxygen ions in the electrolyte lost on the anode side, with a net flow of current through the electrolyte and in the external circuit.

38

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

In an SC-SOFC, the chemical (or electrical) potential gradient across the cell, that is the driving force for the electrical current, is generated by the oxygen partial pressure gradient due to the selectivity of the electrode catalysts for different catalytic reactions, rather than by physically separating the fuel and oxygen as in the conventional dual-chamber design. As a consequence, the need to maintain gas-tight anode and cathode chambers is eliminated and the fuel cell design is greatly simplified. In fact, the SCSOFC not only gets rid of the sealing but even allows for a porous electrolyte. This novel type of SOFC with porous electrolyte is called as fully porous SC-SOFC. Recent studies showed that SC-SOFC with porous electrolytes could also deliver high performances. This allows relatively low-processing temperature of the electrolyte and thus reduces the manufacturing cost. The need for selective electrocatalysts has several implications for the design of an SCFC. First of all, an SC-SOFC must operate at a temperature low enough that the catalysts maintain some degree of selectivity; this typically limits the temperature to below 700°C, which is significantly lower than that of conventional SOFCs with a YSZ electrolyte. For this reason, SC-SOFCs demonstrated to date have used ceria-based electrolytes, rather than YSZ. The relatively low temperatures (400°C 600°C) at which the most advanced SC-SOFCs function also help to ease complications with on-off cycling. The reduced temperatures of operation provide additional benefits, including expanding the choices of materials for fabrication of peripheral components and inhibiting carbon deposition via hydrocarbon cracking at the anode catalyst. Meanwhile, the low temperature also reduces the possibility of explosion of the fuel air mixture. For example, the ignition temperature of methane is higher than 1000°C, especially over Ni and Pt surfaces, and therefore SC-SOFCs with methane fuel can avoid this problem when operated in the reduced temperature range. Another implication of the need for selective electrocatalysts is that an SC-SOFC is unlikely to run well, if at all, on hydrogen. Any catalyst that promotes electrochemical oxidation of hydrogen, or electrochemical reduction of oxygen, would very likely promote direct catalytic combustion if exposed to a hydrogen/air mixture. This problem can be dealt with by using a hydrocarbon fuel instead of hydrogen, as has been done in all successful demonstrations of SC-SOFC operation. With a hydrocarbon fuel, catalytic partial oxidation and reforming chemistry can be used within the anode to deplete incoming oxygen, creating a reducing environment deep within the anode near the electrochemically active layer,

Classification of solid oxide fuel cells

39

and to generate hydrogen needed for the electrochemistry in situ, very near where it is consumed in the electrochemical oxidation reaction. Similarly, at the cathode, if hydrogen generation via hydrocarbon cracking can be suppressed, parasitic combustion at the cathode may be minimized. Fig. 2.12 schematically depicts a SC-SOFC design versus a DC-SOFC design. Compared with DC-SOFCs, the low efficiencies of SC-SOFCs are primarily due to the inherently different flow geometry in which not only half of the fuel passes through the cathode side unreacted, but also the residence time of the flow over the cell is much shorter. In addition, while the usage of large amounts of balance gas in order to prevent explosions in the gas mixture is necessary, it dilutes the fuel stream at the same time and thus lowers the performance. In the experimental studies reported so far, a significant portion utilized un-optimized gas chamber designs and fixed flow rates. Most of these studies tested a single cell instead of a cell stack. Although these test conditions helped to improve single-cell performance and eased both measurement and analysis, they were also, undoubtedly, the major barriers to higher fuel cell efficiencies; keeping SC-SOFCs is just a laboratory curio, insufficient to satisfy real application needs. No need of strict separation of fuel and oxidant in SC-SOFCs is a surprising point that makes it possible to consider various PEN configurations. Fig. 2.13 depicts some typical configurations of SC-SOFCs. SC-SOFCs compared to conventional SOFCs bring some advantages including lower weight as well as smaller volume (specially in planar type) and also needless to complex sealants. Thus, its manufacturing process is much more cost-effective. However, the presence of “spectator” species at the two functional layers of SC-SOFCs (e.g., transport of hydrogen to the cathode functional layer) leads to very low performance compared to Cathode

Cathode

Air Electrolyte

Fuel + air

Electrolyte

Fuel

Anode

Anode (A)

(B)

Figure 2.12 Schematics of (A) dual-chamber versus (B) single-chamber SOFC.

40

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Cathode

Anode

Cathode

Anode

Electrolyte

Electrolyte

(B)

(A) Cathode

Cathode

Electrolyte Electrolyte

Anode

Anode

(D)

(C) Cathode

Cathode

Electrolyte Electrolyte Electrolyte

Anode

(E)

Anode

(F)

Figure 2.13 Schematics of some typical SC-SOFC cell configurations: (A) Co-planar electrodes SC-SOFC, (B) perpendicular electrolyte-supported SC-SOFC, (C) planar electrolyte-supported SC-SOFC, (D) planar anode-supported SC-SOFC, (E) planar cathode-supported SC-SOFC, and (F) planar anode-supported fully-porous SC-SOFC.

conventional SOFCs. Guo et al. [30] successfully designed a novel concept of dual-chamber SOFC technology with porous electrolyte named all porous SOFC (AP-SOFC) which is a bridging technology between DC-SOFC and DC-SOFC. Their cell used methane fuel without any external reformer or steam addition. The porous structure of electrolyte allows gaseous species transport through electrolyte. So, the crack presence and sealant need are not the concern in AP-SOFCs and fuel-O2 management is performed easily to keep the system apart from the explosion risk.

Classification of solid oxide fuel cells

41

2.5.3 No-chamber solid oxide fuel cell As explained, a SC-SOFC design needs neither separator nor gas-seal structure and accordingly can be remarkably simplified. To avoid an explosion of the mixed fuel-air, however, a fuel concentration in the single gas chamber should be rigidly kept higher than its upper explosion limit, for example 15 vol.% for methane. In this safe range, oxygen is deficient for complete oxidation of the fuel. This remains a practical difficulty to manage. Michio Horiuchi et al. [31] reported a fuel cell that generates electricity directly from combustion flame. Since in this unique type of SOFC there is no any chambers, it is well-known as no-chamber SOFC. It is also titled as direct-flame SOFC (DF-SOFC) because the cell exposed is subjected to a direct-flame of a combustion. The operation principle of a direct-flame SOFC (DF-SOFC) is based on the combination of a combustion flame with a SOFC in a simple, “no-chamber” setup illustrated in Fig. 2.14. In this system, a fuel-rich flame is placed at few millimeters from the anode. It serves as partial oxidation reformer while at the same time providing the heat required for SOFC operation. The cathode is freely exposed to ambient air. Flame and fuel cell are geometrically and electrochemically coupled. There are a number of advantages to this approach. First, the system is very fuel-flexible. Because intermediate flame species are similar for all kinds of hydrocarbons, the DF-SOFC can be operated on virtually any carbon-based fuel, as well as other fuels that contain hydrogen. Second, Air

Cathode Electrolyte Anode

Cathode Electrolyte

Secondary flame front: partial oxidation to H2O, CO2

Anode

Partial oxidation products: H2,CO Primary flame front: combustion chemistry and heat release

Flat-flame burner

Bunsen burner

(A)

(B)

Figure 2.14 Operation principle of a direct-flame solid oxide fuel cell: (A) Flat-flame burner and (B) Bunsen-type burner.

42

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

the DF-SOFC is operated in a very simple, no-chamber setup. The anode is simply held into the exhaust gases close to a fuel-rich flame. The cathode breathes ambient air. The system is thermally self-sustained, and there are no high-temperature sealing issues. Third, the system is started up rapidly (i.e., within seconds). The flame heat release brings the fuel cell rapidly to its operation temperature, and there is no external heater required for start-up. These features make the DF-SOFC an attractive system for energy conversion, in particular for combined heat and power applications. There are also a number of drawbacks associated with the DF-SOFCtype setup. This includes the relatively low overall electrical efficiency. An inherent property of the DF-SOFC is that a part of the fuel’s chemical energy is consumed in the combustion reaction and is therefore not available to electrical power generation. Furthermore, material stresses are a particular challenge. The operating environment of a combustion flame can induce significant thermal stress to the SOFC. The DF-SOFC concept is somewhat familiar to the SC-SOFC concept. In the latter, the same premixed fuel/air mixture is supplied to both anode and cathode, and electrochemical fuel oxidation and oxygen reduction is achieved through selective (electro-) catalysts. Within the anode, an H2/CO-rich atmosphere is formed via heterogeneous fuel partial oxidation reactions. In the DF-SOFC, although the setup is even simpler, the two electrodes see different gas atmospheres. The fuel is partially oxidized by homogeneous combustion flame chemistry several millimeters in front of the anode, while the cathode breathes ambient air. This setup relaxes the catalyst selectivity requirement needed for SC-SOFCs and can therefore operate at higher temperature; it also allows higher concentrations of H2/CO at the anode and O2 at the cathode. Thus, the DFSOFC system potentially yields increased performance and efficiency. Furthermore, it does not require an external heater for the start-up phase. Finally, because partial oxidation takes place in the gas phase instead of inside the porous electrodes, the coking problems associated with higher hydrocarbons are significantly reduced in the DF-SOFC compared to the SC-SOFC concept.

2.6 Single and stack cell designs The method of combining the cells, either in parallel or in series, is called stacking. The power density (in the mW/cm2 range) obtained from a

Classification of solid oxide fuel cells

43

single cell is not enough to be practical for a real application. Although this power could be sufficient for milli or micro power range applications, it is still too low for a little larger load application. In order to meet this requirement, several cells must be stacked together either in series or in parallel, and this task can be achieved by interconnecting several cells. Both the planar and the tubular designs have evolved from a single cell and the concept of producing a stack has been problematic. The tubular design offers the possibility of creating gas seals outside of the functional high-temperature zone, but power densities are relatively low. Planar stacks have the advantage of high-power densities, but difficulties associated with sealing and manifolding issues are yet to be overcome. Honeycomb stack design (Fig. 2.15) offers the energy density of planar stacks with the possibility of cold gas seals from tubular design. The main problem is that of designing cells which can be stacked to produce significant power output. This power output is directly proportional to the cell area, so the maximum area of YSZ membrane must be packed into the SOFC reactor stack. This is similar to a heat exchanger design exercise. Two plausible solutions are obvious: a stack of flat plates or an array of parallel tubes. The same heat exchanger problems of joining, cracking, and leakage are evident in the SOFC stacks because of the complex materials and the high-expansion coefficient. Of course, the Anode

Cathode

Electrolyte (middle)

Figure 2.15 Honeycomb-type SOFC.

44

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

+ Cathode bus

Fuel Air

Air

Air Electrolyte

Interconnect

Anode

Air

Air

Air

Cathode

- Anode bus

(A) Upper interconnect Anode Electrolyte Cathode

Fuel Air

(B)

Lower interconnect

Figure 2.16 A schematic of cell configuration in (A) planar and (B) tubular SOFC stack.

difficulties are greater because of the high temperature of operation, highexpansion coefficients, and electrical connection short-circuits. Additional difficulties arise because of the low toughness of the ceramic components and the necessity of making electrical connections between all the cells. Planar cells have the advantage that they can be readily electroded by screen printing, they can be stacked together with narrow channels to

Classification of solid oxide fuel cells

45

achieve high-power densities and they can provide short current pathways through the interconnect. In a tubular stack packed in a square array, as in the Westinghouse design, the power density depends on the diameter of the cells and the gap between them. This is lower than the planar stack because of the relatively large diameter of the tubes. Obviously, HPD depends on having small diameters and less gaps. The microtubular design gives six times better power density at 0.15 cm diameter of electrolyte tube with 0.1 cm spacing [18]. Fig. 2.16 depicts a schematic of tubular and planar SOFC stack configurations.

References [1] J. Weissbart, R.J. Ruka, Oxygen gauge, Rev. Sci. Instr. 32 (1961) 593 595. [2] R.J. Ruka, J. Weissnart, ‘A solid electrolyte fuel cell’, US patent 3,400,054, filed on July 24, 1961. [3] A. Isenberg, W. Pabst, G. Sandstede, Oxygdisches Kathodenmaterial für galvaniche Brennstoffzellen für hohe Temperaturen, DE-P1 571991, filed on October 22, 1966. [4] D.B. Meadowcroft, Properties of strontium-doped lanthanum chromite, Br. J. Appl. Phys. 2 (1969) 1225 1233. Ser. 2. [5] H.S. Spacil, ‘Electrical device including nickel-containing stabilized zirconia electrode’, US patent 3,503,809, filed on October 30, 1964. [6] S. de Souza, S.J. Visco, L.C. De Jonghe, Thin-film solid oxide fuel cell with high performance at low-temperature, Solid. State Ion. 98 (1997) 57 61. [7] J.B. Goodenough, J.E. Ruiz-Diaz, Y.S. Zhen, Oxide-ion conduction in Ba2In2O5 and Ba3In2MO8 (M 5 Ce, Hf, or Zr), Solid. State Ion. 44 (1990) 21 31. [8] T. Ishihara, H. Matsuda, Y. Takita, Doped LaGaO3 perovskite type oxide as a new oxide ionic conductor, J. Am. Chem. Soc. 116 (1994) 3801 3803. [9] M. Feng, J.B. Goodenough, A superior oxide-ion electrolyte, Eur. J. Solid. State Inorg. Chem. T31 (1994) 663 672. [10] K. Huang, R. Tichy, J.B. Goodenough, Superior perovskite oxide-ion conductor strontium- and magnesium-doped LaGaO3: I, Phase relationship and electrical properties, J. Am. Ceram. Soc. 81 (1998) 2565 2575. [11] K. Huang, R. Tichy, J.G. Goodenough, Superior perovskite oxide-ion conductor strontium- and magnesium-doped LaGaO3: II, AC impedance spectroscopy, J. Am. Ceram. Soc. 81 (1998) 2576 2580. [12] National Museum of American History, Behring Center, Smithsonian Institution 2001. [13] US Department of Energy, Office of Fossil Energy. February 2002. [14] Ch.K. Lin, T.T. Chen, Y.P. Chyou, L.K. Chiang, Thermal stress analysis of a planar SOFC stack, J. Power Sources 164 (2007) 238 251. [15] R.K. Shah, A.L. London, Laminar Flow Forced Convection in Ducts, Academic Press, London, UK, 1978. [16] S.C. Singhal, Solid oxide fuel cells for stationary, mobile, and military application, Solid. State Ion. 152-153 (2002) 405 410. [17] K. Huang, S.C. Singhal, Cathode-supported tubular solid oxide fuel cell technology: A critical review, J. Power Sources 237 (2013) 84 97.

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

[18] K. Kendall, M. Kendall, High-temperature solid oxide fuel cells for the 21st century fundamentals, Design and Applications, first ed., Elsevier, 2016. [19] D. Mogensen, J.D. Grunwaldt, P.V. Hendriksen, K. Dam-Johansen, J.U. Nielsen, Internal steam reforming in solid oxide fuel cells: status and opportunities of kinetic studies and their impact on modelling, J. Power Sources 196 (2011) 25 38. [20] P. Aguiar, C.S. Adjiman, N.P. Brandon, Anode-supported intermediate temperature direct internal reforming solid oxide fuel cell. I: model-based steady-state performance, J. Power Sources 138 (2004) 120 136. [21] M. Kamvar, M. Ghassemi, R. Steinberger-Wilckens, The numerical investigation of a planar single chamber solid oxide fuel cell performance with a focus on the support types, Int. J. Hydrog. Energy 45 (2020) 7077 7087. [22] S.P. Jiang, J.G. Love, L. Apateanu, Effect of contact between electrode and current collector on the performance of solid oxide fuel cells, Solid. State Ion. 160 (2003) 15 26. [23] Z. Lin, J.W. Stevenson, M.A. Khaleel, The effect of interconnect rib size on the fuel cell concentration polarization in planar SOFCs, J. Power Sources 117 (2003) 92 97. [24] S. Liu, C. Song, Z. Lin, The effects of the interconnect rib contact resistance on the performance of planar solid oxide fuel cell stack and the rib design optimization, J. Power Sources 183 (2008) 214 225. [25] Q. Chen, Q. Wang, J. Zhang, J. Yuan, Effect of bi-layer interconnector design on mass transfer performance in porous anode of solid oxide fuel cells, Int. J. Heat. Mass. Transf. 54 (2011) 1994 2003. [26] L. Andreassi, G. Rubeo, S. Ubertini, P. Lunghi, R. Bove, Experimental and numerical analysis of a radial flow solid oxide fuel cell, Int. J. Hydrog. Energy 32 (2007) 4559 4574. [27] J. Shi, X. Xue, CFD analysis of a novel symmetrical planar SOFC design with micro-flow channels, J. Chem. Eng. 163 (2010) 119 125. [28] W. Kong, Zh Han, S. Lu, X. Gao, X. Wang, A novel interconnector design of SOFC, Int. J. Hydrog. Energy (2019) (in press). [29] T. Hibino, H. Iwahara, Simplification of solid oxide fuel cell system using partial oxidation of methane, Chem. Lett. 22 (1993) 1131 1134. [30] Y. Guo, M. Bessaa, S. Aquado, M. Cesar Steil, D. Rembelski, M. Rieu, et al., An all porous solid oxide fuel cell (SOFC): a bridging technology between dual and single chamber SOFCs, Energy Environ. Sci. 6 (2013) 2119 2123. [31] M. Horiuchi, S. Suganuma, M. Watanabe, Electrochemical power generation directly from combustion flame of gases, liquids, and solids, J. Electrochem. Soc. 151 (2004) A1402 A1405.

CHAPTER 3

Solid oxide fuel cells in hybrid systems Contents 3.1 Strategies for improving the efficiency of solid oxide fuel cell power generation systems 3.2 Thermodynamic cycle options in hybrid solid oxide fuel cell systems 3.3 Balance of plant equipment 3.3.1 Fuel desulfurization 3.3.2 Heat exchangers 3.3.3 Ejectors 3.3.4 Reformer 3.3.5 Afterburners 3.3.6 Power electronics 3.3.7 Other components 3.4 Basic solid oxide fuel cell/gas turbine hybrid cycle 3.5 Different configurations of solid oxide fuel cell hybrid systems 3.5.1 Direct thermal coupling scheme 3.5.2 Indirect thermal coupling scheme 3.5.3 Other types of coupling 3.6 Mathematical modeling of an solid oxide fuel cell/gas turbine hybrid system References

48 49 51 51 52 53 54 55 55 56 56 56 58 61 63 65 73

A solid oxide fuel cell (SOFC)
based power generation system generates electricity at a high efficiency, and it consequently emits lower CO2 levels when compared with conventional power generation technologies. An SOFC system also emits practically no gaseous pollutants such as CO, NOx, and SOx, and no particulate matter. Therefore, SOFC systems can be regarded as one of the most efficient and clean power generation technologies, particularly in the field of decentralized power generation. In this chapter, a hybrid system, a combination of an SOFC and an internal combustion engine, is proposed and the thermodynamic, economic, and environmental performances of the system is described using

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells DOI: https://doi.org/10.1016/B978-0-12-815753-4.00003-8

© 2020 Elsevier Inc. All rights reserved.

47

48

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

exergy-based methods: exergetic, exergo-economic, and exergoenvironmental analyses. Through the exergetic analysis, the location, magnitude, and sources of thermodynamic inefficiencies in the SOFC power generation systems are identified; chemical reactors and heat exchangers are the main sources of the exergy destruction; however, contrary to common expectation, very little exergy is destroyed within the SOFC stack. This is because part of the generated heat is effectively reutilized within the SOFC stack for internal reforming. An exergo-economic analysis is an appropriate combination of an exergetic analysis and an economic analysis; through this analysis, the information on the cost of the SOFC power generation system is assessed in terms not only of capital investment but also of the cost of exergy. In spite of the progress of SOFC technology, achieved during the recent decades, several barriers still prevent the full commercialization of SOFC systems: firstly, performance degradation needs to be further minimized; secondly, robust operation needs to be guaranteed; and finally, and most importantly, the cost needs to be reduced not only from the manufacturing but also from the operational standpoint. In addition, the SOFC system should be environmentally friendly during its entire life-cycle, that is from cradle to grave. Among the various ways of improving efficiency, economics, and environmental characteristics of the SOFC system, hybridization of SOFC with other power units is introduced in this chapter.

3.1 Strategies for improving the efficiency of solid oxide fuel cell power generation systems In spite of considerable progress, SOFC systems still need to be improved in terms of reliability and cost for full commercialization. Four representative strategies exist: (1) developing new high-performance materials for SOFC, (2) developing high-efficiency balance of plant (BoP) components and consequently reducing parasitic losses, (3) optimizing the cycle structure and optimizing the operational strategies, and (4) increasing efficiency through hybrid. More recently, as fuel cell systems move closer to the commercialization stage, the BoP has received more attention; many studies have been performed focusing on reducing parasitic power consumption, on improving reliability, as well as on reducing cost. Cycle optimization and control algorithm development have also been carried out.

49

Solid oxide fuel cells in hybrid systems

Among the ideas of improving efficiency, SOFC in hybrid systems (known also as hybrid systems) are recognized as very attractive, where other types of power generation technology are combined with a fuel cell, generating extra electric power utilizing unused heat and/or unused fuel. As mentioned in the previous chapters, the operating temperature of SOFC is high, in the range between 500°C and 900°C. Thanks to this high-operating temperature, SOFCs are suitable to deliver waste off-heat to other types of power generation units. In this sense, it can be stated that the SOFC is the most versatile type of fuel cell for hybrid systems.

3.2 Thermodynamic cycle options in hybrid solid oxide fuel cell systems To take advantage of the fuel cell operating parameters, gas and/or steam cycles should be employed. Generally, there are three possibilities (or configurations) considered for the SOFC-based system: 1. the Brayton (gas) regenerative cycle 2. the Rankine (steam) cycle 3. the combined Brayton
Rankine cycle To understand the differences between both cycles and the advantage of combining them, it is important to see their thermodynamics and what parameters characterize them. Such a comparison can be done easily using temperature-entropy (T-s) plots. Although Fig. 3.1 does not have a scale, theoretically, the Brayton cycle operates at higher temperatures. It is reflected in the turbine inlet temperature (TIT), that is the gas turbine always has a higher TIT than the steam Rankine cycle

Brayton cycle 5

Absolute temperature

4 Fuel cell Internal fuel cell HX

Turbine expander

3 6 HRSG

2

Compressor

7

Absolute temperature

Combustor

E

n co

Pump

om

2

ize

Evaporator

r

3

4 Condenser

1

1 Entropy (air)

Entropy (H2O)

(A)

(B)

Figure 3.1 Thermodynamic Brayton and Rankine cycles.

Su

pe

rh

ea

te

r

5 Steam turbine

6

50

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

turbine. In contrast, in a Rankine cycle, evaporation, superheating, expansion in the turbine, and condensation take place at lower temperatures. When going into details, it can be observed that the Brayton cycle heat rejection takes place between points 6 and 7 in the plot. For the Rankine cycle, heat is delivered between points II and V (where III to IV is just evaporation). Employing both cycles in one system allows taking the advantage of the high heat rejection temperature of the gas cycle, and the high heat delivery temperature for the steam cycle. The described idea leads to an increase of the overall system efficiency, but on the other hand the complexity is also higher (i.e., more components involved, complex system, reliability issues). Each cycle has a number of advantages, and some disadvantages when being employed in a hybrid system with a high temperature fuel cell. In the following, the advantages and disadvantages of these three above-mentioned considerations for SOFC-based systems are represented individually. In regenerative Brayton cycles, • the cycle arrangement is simple and a minimum number of components is used; • compressor and turbine pressure ratios are relatively low; • the fuel cell operating pressure is relatively low, hence the problems caused by the anode/cathode pressure differential and high-pressure housing as well as piping are avoided; • gaseous species enter the turbine with relatively low temperature (about 1065°C for the SOFC) thus turbine rotor blades may not require any cooling; • no internal heat transfer surface required for heat removal; • fuel conversion in cells is maximized and the full advantage of fuel cell is taken into account; • the concept is more adaptable to small-scale power generation systems; and • industrial compressor and turbine equipment can be adapted for this application. However, • the operating conditions of the compressor and the turbine equipment are limited to the fuel cell temperature and cycle operating pressure; • a large gas-to-gas heat exchanger is needed for the case of high temperature heat recuperation; and • the efficiency and work output of the cycle is related to the compressor and the turbine efficiencies, pressure losses, and temperature differentials.

Solid oxide fuel cells in hybrid systems

• • • • • • • • • • • • • •

51

In the Rankine cycle, the fuel cell is operated under ambient pressure condition; heat is recovered in a boiler while a high temperature gas-to-gas heat exchanger is required in the regenerative Brayton cycle; there is no need for a compressor; fans for maintaining air and exhaust product gas flow will be sufficient; and steam is available for hybrid applications requiring heat. However, the Rankine cycle inherently has a lower efficiency as compared to the regenerative Brayton and combined Brayton
Rankine cycles; cooling and feed water are required; and the cycle arrangement is more complicated compared to the regenerative Brayton cycle arrangement. In a combined Brayton
Rankine cycle, integrated plant and equipment are available for adaptation to fuel cell heat recovery; and the system is highly efficient in heat recovery. However, these systems are complex, and require multicomponent and largescale components for heat recovery; an adaptation of existing gas turbine is required to provide for air takeoff and return of the hot depleted air and partially burned fuel; high-pressure operation of the fuel cell system is required; precise balancing of the anode and the cathode operating pressures are required to prevent rupture of the fuel cell electrolyte; and indirect heat removal is required from the fuel cell stack by compressed air, initially at low temperature to enable significant conversion of the fuel flow in the cells.

3.3 Balance of plant equipment 3.3.1 Fuel desulfurization The most used nickel anode catalyst is very sensitive to sulfur and its compounds. The maximum tolerable concentration in the fuel is about 0.1 ppm [1]. If no sulfur-free natural gas is available, the fuel has to be desulfurized before entering the SOFC. The usual approach is to break down any organic sulfur compounds to hydrosulfide (H2S) by a hydrogenation catalyst, and subsequently absorb H2S using, for example, zinc oxide or activated carbon. Fixed bed zinc

52

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

oxide absorbers are straightforward, using small cylindrical extrudates of zinc oxide and are designed for operation of several months. Two such cartridges are connected in parallel, so that if one is spent, the other will take over and allow the first to be replaced. Nevertheless, solid-based filter systems such as those using zinc oxide and activated carbon do not possess any sensors to determine the level of exhaustion of the filter cartridges. Therefore, minimum service intervals have to be prescribed that avoid any sulfur “breakthrough.” This leads to conservative estimates of service time, causing exaggerated costs for materials and service personnel.

3.3.2 Heat exchangers Heat exchangers are used extensively in the energy and process industry. In power cycles they are called recuperators and their purpose is to recover heat from exhaust streams for preheating the process streams and therewith saving considerable amounts of fuel. SOFC systems in particular involve elaborate recuperation of heat due to the necessity of reducing thermal stress at the gas inlets, by heating the incoming gas to approximately the stack temperature, and the high amount of heat in the exhaust being processed by gas-to-gas heat exchangers. As explained in Chapter 2, Classification of Solid Oxide Fuel Cells, there are different designs and flow configurations of SOFC, depending on the application. The main design options are tubular and planar types. The choice of design will depend on many parameters, such as flow phase, pressure difference, and available space. Planar heat exchanger types such as the welded or extruded plate-fin or corrugated sheet-based primary surface heat exchangers yield high-heat transfer surface per volume, whereas tube-shell configurations are more robust against pressure differences. Analogous to the fuel cell, flow configurations may be co-flow, counter-flow, or cross-flow. Exergetically, counter-flow is most efficient, because the cold fluid outlet may closely approach the hot fluid inlet temperature if the flow rates and heat exchange surface are suitably chosen. A co-flow configuration may be more effective for heat exchangers with a huge temperature difference between hot and cold fluid and only small temperature changes. Cross-flow in planar heat exchangers is a mixture between the two configurations and easier to design because of the separate inlets and outlets. Attention has to be given to the materials of the heat exchangers. High-temperature operation requires specific steel materials that are less

Solid oxide fuel cells in hybrid systems

53

sensitive to high-temperature corrosion. These will generally be stainless steels which contain 17%
22% of chromium, or alumina formers. The chromium and aluminum content supports the formation of a passivation layer of chromia (chromium oxide) or alumina (aluminum oxide) that will prevent further oxidation and stabilize the surfaces. Nevertheless, chromium from the chromia scale will be volatilized in the presence of water and lead to poisoning of the SOFC cathode. SOFC interconnect design and materials choice takes this into account. In the design of BoP components, the prevention of chromium release needs to be likewise considered, with alumina formers being a current best choice [2].

3.3.3 Ejectors Ejectors, or jet pumps, provide “passive” transport of a gas by mixing it with a gas stream of higher velocity. No moving parts are required, and therefore ejectors may be advantageous when compared to standard compressors operating under extreme conditions. In SOFC systems, ejectors are used to recycle part of the anode exhaust gas in order to supply steam for the reforming reaction. The pressure of the recycled, “induced” fluid has to be lifted slightly to overcome the pressure drop in the fuel cell. The compression energy is supplied from the higher pressure of the “actuating” fresh fuel. High temperatures and chemical aggressiveness of the anode exhaust gases restrict the use of moving parts in blowers and valves, since no moving parts are subjected to elevated temperatures. Regularly, this would involve ball bearings operated at temperatures higher than the evaporation temperature of the lubricants. Therefore, traces of lubricants and wear debris in the gases could damage the anode. The actuating fluid expands through a Laval nozzle and enters the mixing chamber at high velocity. The actuating fluid actually expands to a slightly lower pressure, and thereby creates a suction effect. The actuating fluid thereby accelerates the low-speed fluid in the mixing chamber. By deceleration of the mixture in a diffuser, the dynamic pressure surplus is recovered as static pressure, which hence increases above the inlet pressure of the induced fluid. The principle of an ejector is sketched in Fig. 3.2. The ejector described in Fig. 3.2 is a subsonic mixing ejector. This type allows for a high ratio of induced to actuating fuel, although at only a low-pressure increase. Another option is supersonic mixing, where a

54

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Supersonic flow

Mixing chamber or Diffus

Actuating fluid (fresh fuel)

(dece

leratio

n)

Induced fluid (recycled fuel)

Figure 3.2 Ejector principle.

narrow throat in the mixing chamber accelerates the induced fluid to supersonic and a shock wave accomplishes a higher pressure increase. However, as the SOFC anode pressure drop is typically very low, subsonic mixing ejectors are better suited for SOFC systems. The dimensions of the ejector and the state of the entering gases determine the mixing ratio of recycled and fresh fuel and therewith inherently also the steam-to-carbon ratio. As too low a steam-to-carbon ratio may result in carbon deposition, ejector performance is crucial for safe operation of an SOFC system. The mixing ratio, respectively pressure increase, of an ejector is determined by its design and cannot be actively controlled, and hence it must be part of the operation strategy to ensure a sufficient steam-to-carbon ratio during all operating conditions.

3.3.4 Reformer In reformers, the incoming hydrocarbons are reformed to carbon monoxide and hydrogen according to Eqs. (3.i) and (3.ii) as Cn H2n12 1 nH2 O2ð2n 1 1ÞH2 1 nCO

(3.i)

CO 1 2H2 O2CO2 1 H2

(3.ii)

respectively, while any unconverted hydrocarbon components are reformed directly at the anode (direct internal reforming). The intention of reforming upstream the SOFC is not only to prevent the latter from strong local temperature gradients and high thermal stresses originating from the endothermic reforming reaction but also to supply a certain amount of hydrogen

Solid oxide fuel cells in hybrid systems

55

directly at the inlet to facilitate the fuel conversion. Reformers are reactors that need to be supplied with external heat in order to achieve the desired degree of reforming. Heat recycling from the SOFC exhaust gas stream is therefore key to high system efficiencies and gives the SOFC system a decisive advantage above low-temperature fuel cell technology in that “waste” heat can thus be converted into chemical (fuel) energy that benefits the SOFC stack. A reforming catalyst and a large internal surface of the reformer are required for acceptable performance. Reformers that are thermally integrated with the stack are called indirect internal reformers.

3.3.5 Afterburners Because only part of the fuel can be oxidized in the SOFC due to the electrochemical impossibility of 100% conversion, an afterburner is necessary to combust the residuals. Depending on the fuel cell design, the combustion may occur directly at the fuel cell exit or in a separate chamber. As the fuel is usually highly diluted, special burner configurations are required for stable combustion. Diffusion burners are normally applied as SOFC afterburners. In hybrid systems, the introduction of additional fresh fuel to the afterburner is a measure to stabilize the flame and at the same time enhance the output heat and thus the subsequent turbine power output. Another alternative is to use a catalytic afterburner.

3.3.6 Power electronics Power electronics are necessary to convert the electric power supplied from SOFC and gas turbine to the required grid voltage and frequency. The SOFC delivers direct current (DC). A turbine system rotating constantly at the grid frequency may be directly coupled to the alternate current (AC) grid using a synchronous generator. This is often applied at large power stations, where the inertia of the rotating mass at the same time provides stabilization of the grid frequency. The grid can be seen as an “electric shaft prolongation” from the generator to the consumers. Smaller turbines must, however, rotate faster and possibly with variable speed. It is hence common to convert the generator high-frequency AC to DC with a rectifier. The product is then combined with the DC power from the SOFC and converted to AC at the required conditions with an inverter. Furthermore, the power electronics carries out the “primary control”, that is it adapts the system power output to the power demand from the

56

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

grid (the “load”). There are two parameters controlling the system: the operating point of the SOFC is controlled by the current drawn by the power electronics, and furthermore the turbine shaft speed is controlled by the power drawn from the generator. In case of off-grid applications, an SOFC/gas turbine (GT) power plant may not be able to follow large and quick changes in power demand. In addition, the GT shaft does not provide sufficient inertia for stabilization of frequency and voltage. For these applications, battery banks or flywheels may be required as energy buffers for transient grid stabilization.

3.3.7 Other components The components introduced are most important for the operation of an SOFC/GT hybrid system. However, for startup and shutdown as well as for emergency operation, several additional components are needed. Appliances required might include the following: • Nitrogen supply system: For startup, shutdown, and emergency shutdown cases, nitrogen must be provided to flush the anode in order to protect it from air influx and consequently oxidation. • Auxiliary steam generator: As long as the system includes anode gas recycling, a steam generator will only be required during startup in order to prevent carbon deposition, if no other DI water supply is foreseen. • Auxiliary air blower and cooler/heat rejector: In case of emergency shutdowns, air must be supplied immediately to cool the cell and protect the stack from overheating.

3.4 Basic solid oxide fuel cell/gas turbine hybrid cycle Fig. 3.3 shows the basic configuration of an SOFC/GT hybrid cycle with the components described in Section 3.3. It is basically a recuperated GT cycle where the combustor has been replaced by an SOFC system. Many variations of this cycle have been reported in the literature. The most important options will be discussed in Section 3.5.

3.5 Different configurations of solid oxide fuel cell hybrid systems Availability, an important thermodynamical term, should be obeyed in hybrid plant systems. Specifically, one system (or cycle) needs to supply

Solid oxide fuel cells in hybrid systems

57

Fuel C: Compressor G: Generator HEX: Heat exchanger HT: High temperature LT: Low temperature T: Turbin

Ejector Anodic recirculation loop

Bumer

Pre reformer

Anode Air Cathode LT-HEX

C

HT-HEX

T

SOFC

G Power electronics

Figure 3.3 Basic hybrid GT cycle based on an SOFC system as the heat source.

enough material or heat to meet the requirements of a second system (or cycle). With regard to availability, the hybrid system could be built to operate via either thermal coupling or fuel coupling. Two thermal coupling schemes have been proposed for the exchange of thermal energy between the SOFC system and the targeted combined cycle. In one scheme, known as direct integration, the working medium of one cycle is directly sent to the other cycle as the working fluid. Under this integration scheme, both cycles operate with the same working fluid and at the same pressure. The other thermal coupling scheme, in which the thermal energy is utilized indirectly through a heat exchanger, is known as indirect integration, whereby the heat source cycle and the heat sink cycle potentially operate with different working fluids and at different pressures. It also protects one system from mis-operation of the other, for example pressure surges in the GT subsystem, or chemically corrosive exhaust components. The fuel coupling scheme is used to configure the integration system of an SOFC either through the hydrogen production cycle or using a fuel reformer.

58

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

3.5.1 Direct thermal coupling scheme Fig. 3.4A shows a schematic diagram of a pressurized SOFC and a gas turbine (Brayton cycle) hybrid, the most typical direct thermally coupled scheme. Fuel is supplied to the anode of the SOFC through an appropriate reforming process. Compressed air is provided to the cathode of the SOFC by an air compressor, after being appropriately preheated. As illustrated in Fig. 3.4B, the ideal T-s diagram of the cycle assuming no pressure loss for the Brayton cycle in association with the configuration in Fig. 3.4A, both the air and fuel temperatures increase within the SOFC by the exothermic electrochemical reactions in the SOFC. Within the combustor (COMB), the temperature of the cathode-exhaust oxygendepleted air increases by burning the unreacted fuel released from the anode. The hot combustion gas is expanded in the turbine, the generated shaft-power drives the compressor and generates additional electric power. As shown in Fig. 3.4B, the SOFC is located at the high-pressure side; the operating pressure of the SOFC therefore has a similar value to that of the gas turbine. According to Fig. 3.4B, the change between states 4 and 5 denotes the addition of heat from the combustor. The combustor is part of the SOFC system, following the approach developed by Siemens
Westinghouse [3]. The nonutilized fuel in the product gases is combusted in the combustor. This configuration serves as a basic building block for a SOFC/GT hybrid system. It is challenging to validate system efficiency due to the lack of experimental data for specific fuel cell systems and the BOP; therefore,

T

Anode

Combustor

Fuel Cathode

4 T

SOFC

SOFC Exhaust 3

HEX

6 Heat exch

2

C

5

T

ange

7

C q 1

s

Air

(A)

(B)

Figure 3.4 Pressurized SOFC and gas turbine hybrid system: (A) system configuration, (B) ideal T-s diagram for a Brayton cycle.

Solid oxide fuel cells in hybrid systems

59

the system efficiencies reported by most simulation studies to date have focused on typical hybrid systems. Based on this scheme, we can develop a more complicated hybrid thermodynamic system consistent with the designed conditions. The air compression process may be separated into two stages with an intercooler mounted between the two compressors [4] to decrease the air inlet temperature of the heat exchanger (HEX), thus improving the utilization of exhaust heat. However, as noted by Williams et al. [4], the inlet air temperature of the SOFC will also decrease, which is undesirable for an SOFC. Therefore, the introduction of an intercooler is only recommended for high-operating pressures, as shown in Yi et al. [5]. Zhang et al. [6] inserted an additional re-heater between the HEX and the SOFC by using part of the combustor exhaust gas as a heat source to maintain the minimum temperature of the SOFC stack. Araki et al. [7] added a low-temperature SOFC before a high-temperature SOFC in serial connection. The simulation results showed that the efficiency of the combined cycle with the twostage SOFC was a little higher than with the single SOFC [7]. Similar to the configuration of the two-stage SOFC integration system, a multistage SOFC combined cycle was proposed in Araki et al. [8]. In this system, a few high-fuel utilization SOFC stages were integrated with a Brayton cycle, specifically for CO2 capture. About 10% improvement of efficiency was potentially achieved for this kind of integrated system, compared to that of the conventional CO2 recovery amine process [8]. Fig. 3.5 shows a configuration of a Cheng cycle combined with an SOFC and Brayton cycle. The Cheng cycle is an exhaust heat recovery cycle for specially optimizing the performance of the Brayton cycle. A heat recovery steam generator (HRSG) is added for producing steam, which is injected into the gas before the turbine. The exhaust heat is utilized in the HRSG, thus increasing the system energy efficiency. Because the turbine is a mass-flow device, combining the steam with the hightemperature gases will increase the total electrical power produced. Part of the steam produced in the HRSG can also be used for steam load. Therefore, the Cheng cycle is a good option for following the electrical loads or steam loads via controlling the steam injection. Simulation results showed that the addition of a Cheng cycle to the pressurized SOFC/ Brayton cycle improved the electrical efficiency by 1%
3% points [9]. The Cheng cycle also decreases the temperature in the interior of the turbine, which permits gaining more efficiency via increasing the temperature of the fuel combustion [10].

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Water Exhaust Fuel

HRSG

Anode

Cathode

SOFC

HEX

C

T

Air Figure 3.5 Pressurized SOFC and gas turbine hybrid system combined with a Cheng cycle. Fuel (optional)

T COMB (optional) W

Exhaust

COMB

G

Heat

Cathode

HEX

ange

exch ) (HEX

Atmospheric P SOFC

SOFC

T

C

Combustor (COMB)

Fuel

Anode

W

T

Combustor (COMB-optional)

REF

C

Exhaust

s

Air

(A)

(B)

Figure 3.6 Atmospheric pressure SOFC and gas turbine hybrid: (A) system configuration, (B) ideal T-s diagram for a Brayton cycle.

Fig. 3.6A shows the schematic diagram of an atmospheric pressure SOFC and a gas turbine hybrid. In this configuration, the reactant air is supplied to the cathode after being expanded in the turbine. The SOFC locates at the low-pressure side, as shown in Fig. 3.6B. A combustor can

Solid oxide fuel cells in hybrid systems

61

be installed optionally at the location between the compressor and turbine. The combustor aims at controlling the TIT by means of burning additional fuel; the gas turbine can operate at its own optimal conditions. As shown in Fig. 3.7, a steam injection can be introduced in the configuration, the additional thermal energy is recovered from the exhaust gas in the form of steam [11]. In order to reduce the fuel supply and the load of heat exchangers, recycles can be used for both the anode side and the cathode side [6]. Fig. 3.8 depicts the conceptual diagram of the SOFC system with recycles.

3.5.2 Indirect thermal coupling scheme In a direct thermal coupling scheme, the operation of an SOFC stack is strongly influenced by the operation of another power unit, mainly due to material transfer between the two. To avoid this mutual interaction, indirect thermal coupling schemes can alternatively be introduced. Fig. 3.9 shows a conceptual diagram of an indirectly thermally coupled SOFC hybrid system. As illustrated, interaction between the SOFC and the other bottoming power units is only associated with heat transfer. For the bottoming power unit, any

W

HRSG

REF Fuel

Steam

Anode

Cathode

SOFC

COMB

Mixed gas

HEX

W

G

C

Air Figure 3.7 Pressurized SOFC and steam injected gas turbine hybrid system.

T

62

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Recycle Fuel

W Anode

REF Cathode

SOFC

COMB Mixed gas

Recycle

HEX

W

G

T

C

Air Figure 3.8 Pressurized SOFC and gas turbine hybrid system with recycles.

Fuel

Anode

REF Cathode

SOFC

COMB

Heat

HEX2 (Heat recovery)

Exhaust HEX1

Rankine cycle (steam) ORC Brayton cycle Refrigeration machine

Air

Figure 3.9 Conceptual diagram of indirectly thermally coupled SOFC hybrid system scheme.

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Solid oxide fuel cells in hybrid systems

kind of thermodynamic cycle can be employed, if the operating temperature of the cycle is matched to the temperature of the heat recovery location; steam-based Rankine cycles, organic Rankine cycles, Brayton cycles, and heat-driven refrigeration cycles can be employed. Concerning the HEX2 in Fig. 3.9, from the exergetic viewpoint, there exists an inherent exergy destruction caused by the temperature difference within HEX2. Therefore, the indirect scheme is less favorable in terms of thermodynamic efficiency, whereas it can be more reliable from operation standpoint. Fig. 3.10 shows an indirectly thermally coupled atmospheric pressure SOFC and a gas turbine hybrid configuration. This configuration is very similar to that in Fig. 3.6A except that there exists no material interaction between the SOFC and the gas turbine. The SOFC and the gas turbine have their own independent air supplies, as well as their own independent operating pressure. The SOFC operates at atmospheric pressure, whereas the gas turbine can operate at its own pressure. (Normally the pressure ratio of a gas turbine varies according to the other conditions.) Similar to the case shown in Fig. 3.6, an optional combustor can be integrated accordingly.

3.5.3 Other types of coupling In addition to the above-described two thermal-coupling schemes, other types of integration of SOFC have been suggested, focusing on the Fuel COMB(optional)

W

Fuel REF

Anode

HEX2

Exhaust

Air

Cathode

HEX1

(for SOFC)

COMB

SOFC

W

G

C

T

Exhaust

Air

(for GT)

Figure 3.10 Indirectly thermally coupled atmospheric pressure SOFC and gas turbine hybrid system.

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

various fuel supply options. This coupling can promote the efficient use of fuels and/or utilization of low-grade fuels like coal and biomass. Fig. 3.11 shows a serial integration of an SOFC and a proton exchange membrane fuel cell (PEMFC) [12,13]. The anode off-gas of the SOFC still contains significant amounts of H2 and CO. It can be used as the fuel of a PEMFC after being further converted into H2. For this integration, a water-gas-shift reactor and a preferential-oxidation reactor are required. In order to supply an SOFC with low-cost fuel, like coal and biomass, a gasification process can be integrated with the SOFC, along with suitable gas cleaning devices. Fig. 3.12 depicts a combination of coal gasification and SOFC/GT/ST hybrid system. Through a series of chemical reactors in the coal gasification part, clean synthetic gas can be provided PEMFC part

SOFC part Air

Water

W

REF

Anode Prox

WGS Cathode

Fuel

SOFC Exhaust

W Anode COMB Air

Cathode PEMFC

Depleted air

Figure 3.11 Integration of an SOFC and a PEMFC.

Coal gasification part

Acid gas removal

Steam Heat recovery

SOFC/GT/ST part

Sweet syngas

Clean syngas

Raw syngas

COMB Raw syngas

Particular removal

Polisher Acid gas

Air

T

C

T

HEX

Tail gas Air

Byproduct sulfur

HRSG

Cathode

O2

Anode

Sulfur recovery

Coal (slurry)

Exhaust

Slag Make up water

Water

SOFC

ST

Pump

Condenser

N2 Waste water

Figure 3.12 Integrated coal gasification and SOFC/GT/ST hybrid system.

Solid oxide fuel cells in hybrid systems

65

to the SOFC part. In general, the operating pressure of a gasifier is higher than that of an SOFC and gas turbine. Therefore, a pressure-recovery turbine is installed at the location between gasification part and SOFC part.

3.6 Mathematical modeling of an solid oxide fuel cell/gas turbine hybrid system As stated, a gas turbine cycle is based on the Brayton cycle, which is a simple series of compression, combustion, and expansion processes. The main components of an SOFC/GT hybrid system are a compressor, a combustor, an SOFC, and a gas turbine. The number of components is not limited to four as the cycle may consist of several compressors and turbines (as expanders). In the Brayton cycle, ambient air is compressed and sent to the combustor. The constant pressure combustion takes place, causing expansion, and the exhaust is sent to the turbine where power is extracted by further expansion to drive the compressor and the generator. Heat exchangers can also be used to preheat the stream entering the combustion chamber. Gas turbines are generally used for power production falling in the range of few hundred kilowatts to several hundred megawatts and offer an electrical efficiency of 20%
40%, respectively. This can be further improved by adding a topping cycle to achieve efficiencies of up to 60%. A gas turbine can be directly or indirectly connected to the SOFC. In an indirect integration, the combustor of the gas turbine is replaced by a heat exchanger in which air from the compressor is heated by the fuel cell exhaust and the SOFC can operate under atmospheric conditions. Although this reduces the sealant requirements in the SOFC stack, the heat exchanger has to operate at very high temperatures and pressure differences. Fig. 3.13 shows a simple direct integration of a solid oxide fuel cell and a gas turbine. As can be seen, the combustion chamber of the gas turbine of the Brayton cycle has been replaced by an SOFC and an afterburner. The pressurized stream from the compressor enters the SOFC. The exhaust from the SOFC moves to the afterburner and the resulting high temperature and pressure exhaust enters the turbine. Moreover, heat exchangers are added after the turbine exhaust to further utilize the waste heat in preheating of the streams entering the SOFC stack. Further, heat exchangers could extract more heat for hybrid schemes where electricity and heat from the hybrid system are utilized. Selecting one of the hybrid system configurations described in the previous section is one of the key steps before designing the hybrid system.

66

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Fuel

Anode

Afterburner

Cathode SOFC

C Air

T Exhaust

Figure 3.13 Gas turbine engine as a bottoming cycle in an SOFC/GT hybrid system.

For the aim of mathematical modeling of an SOFC/GT hybrid system, it is necessary to derive the governing equations for each component individually. In the case of an SOFC, the modeling procedure is more complicated, because it operates based on multiphysics principles. This means that several complex processes occur in an SOFC simultaneously while these “physics” are fully coupled to each other. For this reason, modeling of an SOFC behavior including mass, momentum, energy, and charge transport physics as well as electrochemical reactions will be discussed in Chapters 4
6 in more detail. Therefore, the mathematical modeling of the other components will be discussed here. For the purpose of the mathematical modeling of an SOFC component, the readers are referred to the above-mentioned chapters. Note that the outcome of solving all governing equations occurring throughout an SOFC is the polarization curve. A sample polarization curve for an SC-SOFC is shown in Fig. 3.14. As can be seen, the plot shows the cell voltage change with current density. It is worth mentioning that this plot’s slop is always negative. It means that increasing the current density of the cell leads to a cell voltage drop. This is the result of an increasing total overpotential, including ohmic, concentration, and activation overpotentials. This plot also plays a key role in model validation purposes. All researchers who work on numerical investigation of cell performance use this plot to calibrate their results and prove that their prediction for cell performance is accurate. Furthermore, since the product of the cell voltage and the current density is equal to the cell power density, this plot helps to obtain the peak of the power density produced by the cell.

67

Solid oxide fuel cells in hybrid systems

50 1 40

V cell [V]

30 0.6

20

W cell [mW/cm2 ]

0.8

0.4

10

0.2

Anode-supported

0 0

0.02

0.04

0.06

0.08

0.1

0 0.12

I [A/cm2]

Figure 3.14 A sample polarization curve obtained for a SC-SOFC [14].

In the following, the mathematical modeling for each individual component will be presented. In the compressor, ambient air is compressed and supplied to the SOFC. The model of the compressor is based on the ideal gas equations and a polytropic compression process. By referring to thermodynamics books, it is known that in a polytropic process the pressure and the specific volume of a gas are related by [15]: Pv n 5 const:

(3.1)

where P is the absolute pressure, v is the specific volume, and n is the polytropic power of the process whose value is between 1 and k in which k is the ratio of the specific heats of the gas or k5

Cp .1 Cv

(3.2)

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

By applying the ideal gas equation of state which is valid at low pressure and high temperature relative to critical pressure and critical temperature of the gas, respectively, we can write [15] Pv 5 RT

(3.3)

where T is the absolute temperature and R is the gas constant which is related to the universal gas constant (Ru) by [15] R5

Ru M

(3.4)

where Ru is the universal gas constant and is equal to 8.314 kJ/kmol  K and M is the molar mass. Combining Eqs. (3.1) and (3.3) yields
ðn21Þ=n P2 T2 5 T1 (3.5) P1 where subscripts “1” and “2” denote inlet and outlet conditions, respectively. In compressor evaluation, the polytropic efficiency, ηpc , is often used. It is also known as small-stage or infinitesimal-stage efficiency. Taking into account some assumptions, the polytropic efficiency of the compressor is obtained by [16] ηpc 5

k21 k n21 n

By substituting Eq. (3.6) in Eq. (3.5) we obtain [17]
ðk21Þ=kηpc P2 T2 5 T1 P1

(3.6)

(3.7)

The real change in enthalpy between outlet and inlet, Δhreal , of the compressor is obtained by
 T2 Δhreal 5 h2 2 h1 5 CP ðT2 2 T1 Þ 5 CP T1 21 (3.8) T1 It is worth mentioning that for applying Eq. (3.8) the specific heat of the gas is considered as temperature independent and its value is equal to the specific heat of the gas at room temperature (i.e., 27°C or 300K). By substituting Eq. (3.7) in Eq. (3.8) we obtain !
ðk21Þ=kηpc P2 Δhreal 5 CP T1 21 (3.9) P1

Solid oxide fuel cells in hybrid systems

69

In the case of an ideal compressor without any irreversibility, the isentropic process occurs as [15] Pv k 5 const:

(3.10)

For this ideal situation, the ideal change in isentropic enthalpy, Δhideal , is obtained as !
ðk21Þ=k P2 21 (3.11) Δhideal 5 CP T1 P1 The adiabatic efficiency of the compressor, ηC , is defined as the work input required to compress a gas in an isentropic process to the real work input: ηC 5

ideal work wideal 5 wreal real work

(3.12)

Since in the compression process the ideal work is always smaller than the real work, this definition of the adiabatic efficiency of the compressor prevents ηC from becoming greater than 100%, which would falsely imply that the real compressors performed better than the isentropic ones. Also notice that the inlet conditions as well as the exit pressure of the gas are the same for both the real and the isentropic compressors. By applying the first law of thermodynamics, that is energy conservation, and neglecting the changes in kinetic and potential energies of the gas being compressed, the work input to an adiabatic compressor becomes equal to the change in enthalpy and Eq. (3.12) for this case becomes
   ðk21Þ=k  ðk21Þ=k P2 CP T1 PP21 21 21 P1 Δhideal
   5  ðk21Þ=kη ηC 5 5 (3.13) ðk21Þ=kηpc pc Δhreal P2 P2 21 21 CP T1 P1 P1 Note that applying the word “adiabatic” before a component means that heat transfer does not occur between that component and its surroundings (i.e., Q 5 0). Applying the first law of thermodynamics for the compressor as a single-stream component, the steady-flow energy balance equation becomes _ 2W _ 5m Q _ ðΔhactual 1 Δke 1 Δpe Þ

(3.14)

70

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

where m _ is the air flow rate in kg  s21, assuming that the heat transfer from the compressor is negligible (Q  0). Potential energy changes are negligible (ΔPe  0). The velocities involved in compressors are usually too low to cause any significant change in the kinetic energy (ΔKe  0). Under these assumptions, the mechanical power consumed by this steadyflow component (W_C ) can be obtained by   W_C  5 mΔh (3.15) _ actual Using Eq. (3.13) and also regarding the transmission efficiency (ηtrans ) from turbine to compressor, Eq. (3.15) can be rewritten as   _ ideal W_C  5 mΔh ηC :ηtrans

(3.16)

In the hybrid system, the turbine is used to drive the compressor and as a secondary electrical power device. The turbine has been modeled in the same way as the compressor following gas turbine equations for a uniform polytropic expansion. Thus, the exhaust temperature for the gas turbine can be obtained by using Eq. (3.7):
ðk21Þ=kηpGT P2 (3.17) T2 5 T1 P1 where subscripts “1” and “2” denote the inlet and outlet conditions, respectively, k is the ratio of specific heats of the gas defined by Eq. (3.2), and ηpGT is the polytropic efficiency of the gas turbine. For a turbine under steady operation, the inlet state of the working fluid and the exhaust pressure are fixed. Therefore, the ideal process for an adiabatic turbine is an isentropic process between the inlet state and the exhaust pressure. The desired output of a turbine is the work produced, and the isentropic efficiency of a gas turbine, ηGT , is defined as the ratio of the actual work output of the turbine to the work output that would be achieved if the process between the inlet state and the exit pressure were isentropic: ηGT 5

real work wreal 5 ideal work wideal

(3.18)

Usually, the changes in kinetic and potential energies associated with a fluid stream flowing through a gas turbine are small relative to the change in enthalpy and can be neglected. Then, the work output

Solid oxide fuel cells in hybrid systems

71

of an adiabatic turbine simply becomes the change in enthalpy, and Eq. (3.18) becomes
   ðk21Þ=kηpGT  ðk21Þ=kηpGT P2 P2 CP T1 P1 21 21 P1 Δhreal
   5  ðk21Þ=k ηGT 5 5 ðk21Þ=k Δhideal P2 21 CP T1 PP21 21 P1 (3.19) _ GT ), can Now, the mechanical power delivered by the gas turbine, (W be calculated as _ GT 5 mη W _ GT Δhideal

(3.20)

And finally, the mechanical power delivered to the generator to produce electricity is obtained by _ Gen 5 W _ GT 2 W_C W

(3.21)

The hybrid system analyzed uses a parallel flow heat exchanger mode which is the simplest type of heat exchanger. In this type, the heat exchanger consists of two concentric pipes of different diameters, as shown in Fig. 3.15, called the double-pipe heat exchanger. One fluid in a double-pipe heat exchanger flows through the smaller pipe while the other fluid flows through the annular space between the two pipes. In a parallel flow (co-flow) heat exchanger, both the hot and cold fluids enter the heat exchanger at the same end and move in the same direction. The heat exchanger is modeled as ΔT1 5 Th;1 2 Tc;1

(3.22)

ΔT2 5 Th;2 2 Tc;2

(3.23)

where ΔT1 and ΔT2 are the stream-to-stream temperature differences in the inlet and outlet sections of the heat exchanger, respectively. The proportionality between the total heat transfer rate q_ and the overall thermal conductance of the heat exchanger surface is q_ 5 UAΔTlm

(3.24)

where ΔTlm is the log-mean temperature difference and defined as ΔTlm 5

ΔT1 2 ΔT2   2 ln ΔT ΔT1

(3.25)

72

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

T

Hot

fluid

Cold

fluid

Cold out

Hot in

Hot out

Cold in

Figure 3.15 The schematic and temperature profile of a parallel flow (co-flow) double-pipe heat exchanger.

and, d_q 5 2 CdT gives the exit stream temperatures once q_ is known; U is the overall heat exchanger heat transfer coefficient; A is the total heat transfer surface. The streams coming out of the fuel cell are mixed with additional fuel and air in the combustor and the high temperature exhaust is sent to the turbine. The following equation models the flow in the combustor: Enthalpy of f uel cell streams 1 enthalpy of additional fuel 5 net enthalpy of the mixture (3.26)

dH 5 CP ðT ÞdT

(3.27)

Solid oxide fuel cells in hybrid systems

Using Eq. (3.27) and the known mixture enthalpy, !
ðk21Þ=k P2 Δhc 5 CP T1 21 P1

73

(3.28)

Then the exhaust temperature is calculated. The model assumes that the complete combustion is valid and there is no NOx formed during the combustion. In this chapter, a hybrid system, a combination of an SOFC and an internal combustion engine, was discussed and the thermodynamic, economic, and environmental performances of the system were described using exergy-based methods. In the next chapters, we will propose all basics governing physics, including mass, momentum, energy, charge transports coupling with electrochemical phenomena occurring within a single SOFC in detail.

References [1] J. Larminie, A. Dicks, Fuel Cell Systems Explained, John Wiley & Sons, UK, 2000. [2] K. Zhang, A. El-Kharouf, J.-E. Hong, R. Steinberger-Wilckens, The effect of aluminium addition on the high-temperature oxidation behaviour and Cr evaporation of aluminised and alumina-forming alloys for SOFC cathode air pre-heaters, Corros. Sci. (2020). in print. [3] R.A. Roberts, J. Brouwer, Dynamic simulation of a pressurized 220 kW solid oxide fuel-cell
gas-turbine hybrid system: modeled performance compared to measured results, ASME J. Fuel Cell Sci. Technol. 3 (2006) 18
25. [4] G.J. Williams, A. Siddle, K. Pointon, Design optimization of a hybrid solid oxide fuel cell & gas turbine power generation system, ALSTOM Power Technology Centre Report, 2001. [5] Y. Yi, A.D. Rao, J. Brouwer, G. Scott Samuelsen, Analysis and optimization of a solid oxide fuel cell and intercooled gas turbine (SOFC
ICGT) hybrid cycle, J. Power Sources 132 (2004) 77
85. [6] X. Zhang, J. Li, G. Li, Z. Feng, Cycle analysis of an integrated solid oxide fuel cell and recuperative gas turbine with an air reheating system, J. Power Sources 164 (2007) 752
760. [7] T. Araki, T. Ohba, S. Takezawa, K. Onda, Y. Sakaki, Cycle analysis of planar SOFC power generation with serial connection of low and high temperature SOFCs, J. Power Sources 158 (2006) 52
59. [8] T. Araki, T. Taniuchi, D. Sunakawa, M. Nagahama, K. Onda, T. Kato, Cycle analysis of low and high H2 utilization SOFC/gas turbine combined cycle for CO2 recovery, J. Power Sources 171 (2007) 464
470. [9] K. Onda, T. Iwanari, N. Miyauchi, K. Ito, T. Ohba, Y. Sakaki, et al., Cycle analysis of combined power generation by planar SOFC and gas turbine considering cell temperature and current density distributions, J. Electrochem. Soc. 150 (2003) A1569
A1576. [10] H. Lee Willis, W.G. Scott, Distributed Power Generation: Planning and Evaluation, CRC Press, 2000.

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

[11] A. Mazzucco, M. Rokni, Thermo-economic analysis of a solid oxide fuel cell and steam injected gas turbine plant integrated with woodchips gasification, Energy 76 (2014) 114
129. [12] H.E. Vollmar, C.U. Maier, C. Nolscher, T. Merklein, M. Poppinger, Innovative concepts for the coproduction of electricity and syngas with solid oxide fuel cells, J. Power Sources 86 (2000) 90
97. [13] K. Subramanyan, U.M. Diwekar, A. Goyal, Multi-objective optimization for hybrid fuel cells power system under uncertainty, J. Power Sources 132 (2004) 99
112. [14] M. Kamvar, M. Ghassemi, R. Steinberger-Wilckens, The numerical investigation of a planar single chamber solid oxide fuel cell performance with a focus on the support types, Int. J. Hydrog. Energy 45 (2020) 7077
7087. [15] Y.A. Cengel, M.A. Boles, M. Kanoglu, Thermodynamics: An Engineering Approach, ninth ed., Mc Graw Hill, 2019. [16] M.P. Boyce, Gas Turbine Engineering Handbook, second ed., Gulf Professional Pub, 2002. [17] P. Chinda, P. Brault, The hybrid solid oxide fuel cell (SOFC) and gas turbine (GT) systems steady state modeling, Int. J. Hydrog. Energy 37 (2012) 9237
9248.

CHAPTER 4

Fundamentals of electrochemistry Contents 4.1 The basic concepts of gas mixture category 4.1.1 Mass fractions and mole fractions 4.1.2 Ideal gas mixtures 4.1.3 Properties of gas mixtures 4.2 Conservation of species 4.3 Species source terms in solid oxide fuel cells 4.3.1 Chemical reactions 4.3.2 Electrochemical reactions 4.3.3 Some applicable boundary conditions for solid oxide fuel cells References Further reading

76 76 77 79 80 87 88 91 96 98 99

As you may know, in all types of fuel cells the multicomponent mass, electron, and ion transport are one set of governing physics in solid oxide fuel cells (SOFCs). The aim of this chapter is to explain the fundamental electrochemistry occurring in SOFC systems. But in order for an electrochemical reaction to occur, the multicomponent mixture should be transferred from the electrode channels toward the near of the electrode
electrolyte interface called the catalyst layer where the electrochemical reactions occur. After that, oxide ions and electrons are produced or consumed and also are transferred. So, to explain the electrochemistry of SOFCs, we should deal with the ion, electron, and multicomponent mass physics. In SOFCs, electrolyte layer should be fully dense in order not to allow fuel and oxidant to be mixed. It should also be insulated against electron passage in order to allow electron to be transferred through external electrical circuit, and electricity is produced. For this reason, mass and electron transport physics is not applicable for this layer. Furthermore, ion and electron transport are disabled for electrode backing layers and channels Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells DOI: https://doi.org/10.1016/B978-0-12-815753-4.00004-X

© 2020 Elsevier Inc. All rights reserved.

75

76

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

because it is assumed that the electrochemical reactions occurrence is within the catalyst layers. In this chapter, as we deal with multicomponent gas mixture especially in mass transport physics it is useful to start the chapter with the basic concepts of gas mixture category. Then the extraction of multicomponent mass transport governing equation in free (channels) and porous (electrodes) is explained. Then the factors that cause mass generation phenomena, including chemical and electrochemical reactions, are explained. Since the species source term of electrochemical reactions is completely related to charge (ion and electron) transport physics, it is described in this subsection. Finally, to fulfill the modeling, the set of applicable boundary conditions are presented.

4.1 The basic concepts of gas mixture category The benefit associated with SOFC high-operating temperature is that different types of hydrocarbon-based fuels such as methane and natural gas can be used without need for external reforming. As known, the combustion of hydrocarbon fuels causes multicomponent stream such as hydrogen, carbon monoxide, and carbon dioxide species at the anode side and oxygen and nitrogen species at the cathode side. Therefore, it is necessary to review some basic concepts that are related to the gas mixtures.

4.1.1 Mass fractions and mole fractions Mass and mole fraction are the two important components of any specific gas mixtures. They are used to describe gas concentrations as well as to determine the vapor pressures of mixtures of similar liquids. Mass fraction (yi), also known as weight fraction, is the ratio of the ith species the mass, mi, to the total mass of the mixture, mtot: mi yi 5 (4.1) mtot The total mass of the mixture, the sum of the individual masses of the different species that existed in the mixture, is equal to one: X yi 5 1 (4.2) i

Similarly, the mole fraction, xi, is the ratio of mole of an ith species in the mixture, ni, to the total mole of the mixture, ntot, as formulated by

77

Fundamentals of electrochemistry

xi 5

ni ntot

(4.3)

where the number of moles of ith species in a mixture, ni, is the ratio of the ith species mass to its molar mass, Mi: mi ni 5 (4.4) Mi Substituting Eq. (4.4) into Eq. (4.3) and considering the fact that the total number of moles of the mixture is the sum of the number of moles of each components that existed in the mixture, the mole fraction equation becomes mi Mi

xi 5 P

mi i Mi

(4.5)

Again, the total number of moles of the mixture, the sum of the number of moles of all species that existed in the mixture, is equal to one: X xi 5 1 (4.6) i

Also, by substituting Eq. (4.4) into Eq. (4.1) the mass fraction equation becomes as follows: yi 5

ni Mi ntot Mmix

(4.7)

When dealing with SOFC it is more convenient to use molar mass for chemical reaction. However, in practice and usually the amount of mass fractions of the components in the mixture are known. The total molar mass of the mixture, Mmix, is calculated by X xi Mi (4.8) Mmix 5 i

Combining the mole fraction relation, Eq. (4.3), and Eq. (4.8) yields xi Mi yi 5 P x i Mi

(4.9)

4.1.2 Ideal gas mixtures Gases in which the interactions between its molecules are neglected are known as ideal gases. In another word, the interaction between moving

78

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

molecules of ideal gases are purely elastic collisions. The basis of ideal gas is statistical thermodynamics, also called equilibrium statistical mechanics. Statistical thermodynamics is used to explain how the theory of macroscopic properties of large systems, such as volume, pressure, and temperature, are related to the concept of their microscopic state. Ideal gas law was originally derived from the experimentally measured Boyle’s law, for low pressure and isothermal process, and Charles law, for high temperature and isobaric process. The Boyle’s law state that in an isothermal process and for a given mass the pressure is inversely proportional to volume (PV 5 F(T)), while the Charles law state that in an isobaric process and for a given mass, the volume is directly proportional to temperature (V/T 5 G(P)). P and V are the absolute pressure and the total volume of the mixture, respectively, and T is the absolute temperature. The relation between pressure, temperature, and volume of any ideal gasses obeys the following equation of state: PV 5 nmix Ru T

(4.10)

where nmix is the total number of mole of the mixture and Ru is the universal gas constant and is equal to 8.314 kJ/kmol  K. Generally, gases that has very low reduced pressure (PR , , 1), the ratio of gas pressure to its critical pressure, or has high reduced temperature (TR . 2), the ratio of gas temperature to its critical temperature, behave as an ideal gas and therefore the ideal gas equation of state, Eq. (4.10), can be used for these gases with reasonable accuracy. To accurately determine whether a gas behaves as ideal or not the compressibility factor, Zmix, in the equation of state of gases, as stated below, should be higher than 96%: PV 5 Zmix nmix Ru T

(4.11)

The mixture compressibility factor, Zmix, is expressed in terms of the compressibility factors of the individual gases, Zi, as follows: Zmix 5

k X

yi Zi

(4.12)

i51

where Zi is determined using generalized compressibility charts given in most thermodynamics books.

Fundamentals of electrochemistry

79

4.1.3 Properties of gas mixtures The density of the mixture, ρmix, the ratio of volume to mass of mixture, is calculated by [1] 1 ρmix 5 P yi

(4.13)

i ρi

where the density of each species, ρmix, is obtained from equation of state of ideal gas when the gas mixture behaves as an ideal mixture [2]: ρi 5

pi Mi Ru T

(4.14)

where pi is the absolute partial pressure of the ith species, Mi is the molar mass of the ith species, Ru is the universal gas constant and is equal to 8.314 kJ/kmol  K, and T is the absolute temperature. Concentration of each components in the mixture, ci, the ratio of mixture density to its molar mass, is obtained by ρ xi p (4.15) ci 5 i 5 RT Mi where xi is the mole fraction of the ith species and p is the total absolute pressure of the mixture. The total concentration of the mixture, cmix, is determined by cmix 5

N X

ci

(4.16)

i51

Other relations needed to describe gas concentration are mixture mass
based internal energy, umix, mixture molar
based internal energy, umix , mixture mass
based enthalpy, hmix, mixture molar
based enthalpy, hmix , mixture mass
based specific heat capacity, Cpmix, mixture molar
based specific heat capacity, and mixture mass
based entropy, Smix. Molar-based entropies are as follows, respectively: umix 5

N X yi ui i51

N X umix 5 xi ui i51

(4.17)

80

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

hmix 5

N X yi hi i51

N X hmix 5 xi h i

(4.18)

i51

cp;mix 5 c p;mix 5

N X

yi cp;i

i51 N X

(4.19)

xi c p;i

i51

smix 5

N X yi si i51

N X smix 5 xi si

(4.20)

i51

4.2 Conservation of species Just like other conservation principal (i.e., conservation of mass, momentum, and energy), the conservation of species principle for a control volume and a time interval Δt, as depicts in Fig. 4.1, state that the net transfer of species from/to a control volume is equal to the net change in the total species within the control volume. A better presentation of the species principle is given in the following form: Using the Taylor series expansion, the mathematical form of the species equation, Eq. (4.21), is as follows:   @ @ ðρi dxdydzÞ5 ðρi ui dydzÞ2 ρi ui 1 ðρi ui Þdx dydz @t @x     @ @ 1ðρi vi dxdzÞ2 ρi vi 1 ðρi vi Þdy dxdz 1 ðρi wi dxdyÞ2 ρi wi 1 ðρi wi Þdz dxdy @y @z 1 S_ s;i dxdydz

(4.22)

where ρi is the density of the species “i”, u, v, and w are the convection or bulk motion velocity components in the x, y, and z directions, respectively. Similarly, Ui, Vi, and Wi are the diffusion velocity components of

Fundamentals of electrochemistry

81

Figure 4.1 The mass flux of species i applied on control volume “dxdydz.”

0

1 0 1   Rate of repletion Rate of production @ or depletion of species i A 5 Net rate of species 1 @ or consumoption A ðIn-outÞ within the CV of species i (4.21)

species “i” in the x, y, and z directions, respectively, and S_ s;i is the rate of consumption or production of species “i” per unit volume. By substituting the absolute velocity of ith species in terms of conservation and diffusion components into Eq. (4.22), it reduces to the species equation also known as species continuity equation as follows:    @ρi @  @ @  ρ ðu 1 Ui Þ 1 ρ ðv 1 Vi Þ 1 ρ ðw 1 Wi Þ 5 S_ s;i 1 @x i @y i @z i @t

(4.23)

The vector notations of the species continuity equation, Eq. (4.23), take the form of the following:   @ρi 1 r: ρi ðv 1 Vi Þ 5 S_ s;i @t

(4.24)

The second term in the left-hand side of the continuity equation, Eq. (4.24), comprises of two flux terms: the convectional and the diffusional fluxes. Typically, the effect of convectional flux term is negligible compared with the diffusional flux term especially when the behavior of

82

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

the SOFCs is the focus of the study. Considering this fact, the species continuity equation, Eq. (4.24), reduces to   @ρi 1 r: ρi Vi 5 S_ s;i @t

(4.25)

The molar diffusional flux model of the conservation of species, Eq. (4.25), is written as @ρi 1 r:½Ni  5 S_ s;i @t

(4.26)

where S_ s;i represents the volumetric production or consumption of species i and Ni is the molar diffusional flux of species i and is explicitly expressed by [3] Ni 5 Ni ðcÞ 1 Ni ðpÞ 1 Ni ðgÞ 1 Ni ðT Þ

(4.27) ðpÞ

where NiðcÞ is the molar concentration (ordinary) diffusion flux, Ni is the ðgÞ molar pressure diffusion flux, Ni is the molar body-force diffusion flux, ðT Þ and Ni is the molar thermal diffusion (Soret effect) flux. Among all the mentioned molar diffusion fluxes, the molar concentration (ordinary) diffusion play an important role in modeling the mass transport of SOFCs. Therefore, other terms are molar diffusion terms, except the molar concentration, are neglected, and the molar diffusion flux, Eq. (4.27), contains only the molar concentration (ordinary) diffusion term as follows: Ni 5 NðcÞ i

(4.28)

The molar concentration diffusion flux is determined using any constitutive models such as the Fick’s law of diffusion, the Stefan
Maxwell model, or the Dusty-gas model (DGM). However, the Stefan
Maxwell model provides the most general and convenient approach to the multicomponent mass transport modeling inside the free and porous layers of the SOFCs. Based on the Stefan
Maxwell model the diffusive flux of each species in multicomponent diffusion with n components depends on the concentration gradient of the other n 2 1 species [4]. More importantly, the Stefan
Maxwell model can be modified to include the effect of Knudsen diffusion, the collision between the species (i.e., the gas molecules), or the collision between the gas molecules and the pore walls of the porous electrodes.

Fundamentals of electrochemistry

83

The Knudsen diffusion effect inside the porous electrode layers is determined by the Knudsen number (Kn), the ratio of the mean free path of gas species to the pore diameter, as follows: Kn 5

λ dp

(4.29)

where λ is the mean free path of the gas molecule and is determined by the kinetic theory of gasses as follows: kB T λ 5 pffiffiffi 2πdp2 p

(4.30)

kB is the Boltzmann constant and is approximately 1.3807 3 10223 joules per kelvin (J  K21 ), T is the absolute temperature, dp is the average pore diameter of molecules, and p is the absolute pressure. The diffusion in porous media is typically divided into three regimes: 1. The continuum regime where the Knudsen number is less than 0.01, Kn , 0.01. 2. The transition regime where the Knudsen number is between 0.01 and 1, 0.01 , Kn , 1. 3. The Knudsen regime where the Knudsen number is greater than 1, Kn . 1. In SOFC electrodes, the ordinary diffusion effect is comparable with the Knudsen diffusion effect. This implies that the diffusion in SOFC electrodes is in transition regime which means the diffusion Knudsen number is between 0.01 and 1. The Stefan
Maxwell equation that applies to multicomponent systems with n species are as follows [5]:  n  X  xi xj  Vi 2 Vj rxi 5 2 Dij j51

(4.31)

The diffusion velocities can be replaced by diffusion fluxes through the following relation: Vi 5

Mi Ni ρi

(4.32)

84

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Using Eqs. (4.5) and (4.16), Eq. (4.33) reduces to Vi 5

Ni xi c

(4.33)

By substituting Eq. (4.33) into Eq. (4.31) the Stefan
Maxwell equation for multicomponent systems with n species becomes rxi 5

n X  1  x i Nj 2 x j Ni cDij j51

(4.34)

where xi is the mole fraction of the species i, c is the concentration of the mixture, and Ni is the diffusion flux of the species i. Dij is the ordinary diffusion coefficient of the species i in j and is determined using two different formulas: Chapman
Enskog formula and Fuller et al. formula [5]. The Chapman
Enskog formula for ordinary diffusion coefficient, Dij, is as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

T 3 M1 i 1 M1 j Dij 5 1:883 3 1027 (4.35) pσ2ij ΩD;ij where T is the absolute temperature in K, p is the total pressure and its unit is atmosphere, M is the molecular mass of the gas and its unit is g/mol, and ΩD,ij is a dimensionless function of temperature and intermolecular potential field and is directly affected by the Boltzmann constant, kB, and temperature, T, and indirectly affected by the molecular energy parameter, ε(kBT/ε). The unit for the ordinary diffusion coefficient, Dij, is in m2/s. The value of σij is estimated as the arithmetic average of the pure component values and is given as follows: σij 5

σi 1 σj 2

(4.36)

The unit for the σij is Å. The Fuller et al. formula for the ordinary diffusion coefficient, Dij, is as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffi 1 1 Mi 1 Mj 27 1:75 (4.37) Dij 5 1:013 3 10 T pffiffiffi pffiffiffiffi p 3 vi 1 3 vj Þ2

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85

The terms vi and vj are molecular diffusion volumes and are available for some simple molecules in Table 4.1. It is also usual to use the Knudsen diffusion. The Knudsen diffusion mechanism accounts for the collisions between the species and the surrounding media. An example of such mechanism is the passage of the species through the electrodes pore walls. The Knudsen diffusion coefficients of i and j species, DKn,ij, is determined by [6] vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 u u 8Ru T

DKn;ij 5 dp t (4.38) 3 π 1 11 Mi

Mj

where dp is the pore diameter, Ru is the universal gas constant and is equal to 8.314 kJ/kmol  K, T is the absolute temperature and its unit is K, and M is the molecular weight of different species and its unit is kg/mol. As known, pores play the role of retarding the transport of mass through the porous medium. The anode and cathode electrodes of the SOFC is a porous media and therefore it is expected to have lesser diffusion coefficients compared with mass transport through free channels. To account for the effect of the mass transport through the porous electrodes of the SOFC, a combined effective diffusion coefficient must be used. eff Two combined effective diffusion coefficients,Dij , that are used are as follows, respectively [6]: ε eff Dij 5 Dij (4.39) τ or Dij 5 ετ Dij eff

(4.40)

ε is the porosity of the porous electrode and is given by ε5

volume of void space total volume

(4.41)

Table 4.1 The molecular diffusion volumes for some simple molecules used in SOFC systems [5].

H2

O2

CO

CO2

H2O

N2

6.12

16.3

18.0

26.7

13.1

18.5

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

And τ is the tortuosity of the porous electrode and is given by τ5

actual path length point-to-point path length

(4.42)

By substituting the combined diffusion coefficient with the effective combined diffusion coefficient in Eq. (4.34), the modified Stefan
Maxwell equations, Eq. (4.34), becomes as follows: rxi 5

n X 1  eff j51 cDij

xi Nj 2 xj Ni



(4.43)

In addition, the DGM is also used to account for both molecular diffusion as well as the Knudsen diffusion. The combined effective binary diffusion coefficient based on the DGM, DijDGM , is as follows: ! eff D 3 D ε Kn;ij ij (4.44) DijDGM 5 τ DKn;ij 1 Dijeff It is common to use the matrix form of the modified Stefan
Maxwell model in numerical application. The matrix form of the modified Stefan
Maxwell relation, Eq. (4.34), is as follows [5]: c ðrxi Þ 5 2 ½BðNi Þ

(4.45)

[B] is a square matrix of order n 2 1 and its elements are extracted by Bii 5

xi eff

Dik

k X xj

1

eff j51;i6¼j Dij

where

(4.46)

! Bij 5 2 xi

1 eff

Dij

2

1 eff

Dik

(4.47)

Multiplying both sides of Eq. (4.45) by the inverse of square matrix [B] gives c ½B21 ðrxi Þ 5 2 ½B21 ½BðNi Þ

(4.48)

Or in terms of the molar diffusional flux of species i (Ni), Eq. (4.48) becomes as follows: ðNi Þ 5 2 c ½B21 ðrxi Þ

(4.49)

Fundamentals of electrochemistry

where the inverse matrix, [B] 2

21

87

, is given as

Γ 11 6 Γ 21 ½B21 5 ½Γ  5 6 4 ^ Γ n21;1

Γ 12 Γ 22 Γ n21;2

3 Γ 1;n21 Γ 2;n21 7 7 5 ^ ? Γ n21n21 ? ?

(4.50)

where Γ is the element of the matrix. The elements of matrix [B] as well as the inverse matrix [B]21 are not constant and depend on the operating and design parameters of the SOFC, such as temperature, pressure, species concentrations, pore size, porosity, and tortuosity. The diffusion fluxes of different species, (Ni), is explicitly expressed as follows:   N1 5 2 c Γ 11 rx1 1 Γ 11 rx1 1 ? 1 Γ 1;n21 rxn21  N2 5 2 c Γ 21 rx1 1 Γ 22 rx2 1 ? 1 Γ 1;n21 rxn21 (4.51) ^   Nn21 5 2 c Γ n21;1 rx1 1 Γ n21;2 rx2 1 ? 1 Γ n21;n21 rxn21 Substituting the diffusion fluxes, Eq. (4.51), in the molar diffusional flux model of the conservation of species, Eq. (4.26) provides the conservation of species in terms of gradients of mole fractions of species i as follows: @ρi 1 r:½ 2c ðΓ i1 rx1 1 Γ i2 rx2 1 ? 1 Γ in rxn Þ 5 S_ s;i @t

(4.52)

4.3 Species source terms in solid oxide fuel cells The species source terms basically originate from the chemical and electrochemical reactions which occurs at the anode and cathode sides of the SOFCs. The species source terms that occur in other layers (i.e., source terms associated with the air and fuel channels as well as the source term associated with the cathode backing layers) are negligible and the Eq. (4.26) for these layers is reduced to @ρi 1 r:½Ni  5 0 @t

(4.53)

The details of how the source/sink terms are generated during the chemical and electrochemical reactions in SOFC are described separately in the following sections.

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

4.3.1 Chemical reactions The two most important chemical reactions that take place in any SOFCs with fuels (such as methane) except hydrogen fuel are steam-reforming reaction and water
gas shift reaction. In some types of fuel cells such as the single-chamber SOFCs, where fuel and oxidant simultaneously are fed to the cell, the combustion chemical reaction is inevitable. The general form of chemical reaction is as follows: kf

aA 1 bB " cC 1 dD

(4.54)

kb

where A and B are called reactants, whereas C and D are called products of the reaction. kf is the forward reaction rate constant, kb is the backward reaction rate constants, and the double arrow, (") pointing in opposite directions, indicate that the reaction is in equilibrium. In addition, there is two-way reaction in most SOFCs, there are some one-way reaction, such as methane combustion, that take place in single-chamber SOFCs. However, the principle associated with the rate of consumption (production) of reactants (products) for all chemical reactions is the same and is determined by Arrhenius equation. According to Arrhenius, the reaction rate constant, k, is formulated by   k 5 Aexp 2 Ea =Ru T

(4.55)

where A is preexponential factor, Ru is the universal gas constant and is equal to 8.314 kJ/kmol  K, T is the absolute temperature and its unit is K, and Ea is the activation energy and its unit is kJ/kmol  In addition, the volumetric rate, r, used in chemical reaction, is determined by r 5 kf paA pbB 2 kb pcC pdD

(4.56)

where the partial pressure of species i in a mixture, pi, from the thermodynamics principle is related to the total pressure of the mixture, p, by p i 5 xi p

(4.57)

By substituting Eq. (4.57) into Eq. (4.56), the volumetric rate in terms of mole fractions is determined and is as follows: r 5 kf xaA xbB pa1b 2 kb xcC xdD pc1d

(4.58)

Finally, the rates of production or consumption, S_ A , of various species that takes place in chemical reaction, Eq. (4.54), is determined by

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89

S_ A 5 2 ar

(4.59)

S_ B 5 2 br

(4.60)

S_ C 5 cr

(4.61)

S_ D 5 dr

(4.62)

The subscripts “A,” “B,” “C,” and “D” denote the species “A,” “B,” “C,” and “D” taking apart in chemical reaction Eq. (4.54). To apply the general form of chemical reaction to SOFC, the methane fuel (as an example) is considered. In this case two important chemical reactions named the methane-reforming reaction and the water
gas shift reactions occur, which are as follows, respectively: • Methane-reforming reaction kbr

CH4 1 H2 O " 3H2 1 CO kfr

•

ΔH298 5 206kJ=mole

(4.i)

Water
gas shift reaction: kfs

CO1H2 O " H2 1CO2 kbs

ΔH298 5 2 41:1kJ=mole

(4.ii)

The volumetric reaction rates for the methane-reforming reaction, rr, and for the water
gas shift reaction, rs, are as follows, respectively: h i rr 5 p2 kfr xCH4 xH2 O 2 p2 kbr x3H2 xCO

(4.63)

  rs 5 p2 kfs xCO xH2 O 2 kbs xH2 xCO2

(4.64)

The empirical relation for the methane-reforming equilibrium constant is Kpr 5

kfr 5 1:0267 3 1010 kbr   3 exp 20:2513ζ 4 1 0:3665ζ 3 1 0:5810ζ 2 2 27:134ζ 1 3:2770 (4.65)

And the empirical relation for water
gas shift reactions equilibrium constant is [7]

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Kps 5

  kfs 5 exp 20:2935ζ 3 1 0:635ζ 2 1 4:1788ζ 1 0:3169 kbs

(4.66)

where ζ is determined by ζ5

1000 21 T ðKÞ

(4.67)

The preexponential factor and energy activation for forward reaction rate constants of the methane-reforming reaction are 2395 mol/m3Pa2s and 231266 J/mol, respectively. Also, the preexponential factor and energy activation for forward reaction rate constants of the water
gas shift reaction are 0.0171 mol/m3Pa2s and 103191 J/mol, respectively. Once the forward reaction rate and equilibrium constants for the methane-reforming and water
gas shift reactions are determined, the backward reaction rate constants are calculated by setting the Eqs. (4.65) and (4.66) equal to each other. Lastly, the production and/or consumption rate for five different species that take part in the anode layer methane-reforming and water
gas shift reaction is determined as follows: Consumption rate of methane due to methane-reforming reaction, S_ CH4 , is calculated by S_ CH4 5 2 rr

(4.68)

The minus sign in Eq. (4.68) means that the methane is consumed due to methane-reforming reaction. Production rate of hydrogen due to methane-reforming reaction and water
gas shift reaction, S_ H2 , is calculated by S_ H2 5 3rr 1 rs

(4.69)

It means that hydrogen is produced by the methane-reforming reaction as well as water
gas shift reaction. Consumption rate of water due to methane-reforming reaction as well as water
gas shift reaction, S_ H2 O , is calculated by S_ H2 O 5 2 rr 2 rs

(4.70)

It means that water is consumed by the methane-reforming reaction as well as water
gas shift reaction. The source term of carbon monoxide can be formulated as S_ CO 5 rr 2 rs

(4.71)

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91

It means that carbon monoxide is produced by methane-reforming reaction but is consumed by water
gas shift reaction. Production rate of carbon dioxide due to methane-reforming reaction, S_ CO2 , is calculated by S_ CO2 5 rs

(4.72)

It shows that carbon monoxide is produced through only water
gas shift reaction.

4.3.2 Electrochemical reactions As explained in Chapter 1, Introduction to fuel cells, the two main electrochemical reactions that take place in any SOFCs are the oxygen reduction in cathode electrode and the hydrogen oxidation in anode electrode. These electrochemical reactions lead to consumption of oxygen at the cathode electrode and hydrogen at the anode electrodes. The governing equations that account for the modeling of the consumption of oxygen and hydrogen are the electric charge transfer and ionic charge transfer. The electric charge and ionic charge conservation statement that is applied to a differential control volume (CV) shown in Fig. 4.2 is written as follows: The transfer of ions and electrons through solid portion of the porous electrodes is very fast and therefore the conservation statement is assumed to be time independent. The mathematical form of the electric and ionic charge conservation statement is as follows: ! @Jx 0 5 ðJx dydzÞ 2 Jx 1 dx dydz @x ! ! @Jy @Jz dy dxdz 1 ðJz dxdyÞ 2 Jz 1 dz dxdy 1 ðJy dxdzÞ 2 Jy 1 @y @z 1 S_ c dxdydz (4.74) Or by simplifying, Eq. (4.74) becomes as @Jy @Jx @Jz _ 1 1 5 Sc @x @y @z

(4.75)

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Figure 4.2 The current density in x and y applied on control volume “dxdydz.” 0 1 0 1   Rate of accumulation Rate of production Net rate of current @ of electric or ionic charge A 5 A 1@ or consumoption ðIn-outÞ within the CV of electric or ionic charge (4.73)

where Jx, Jy, and Jz are the current densities in the x, y, and z directions, respectively, and S_ c is the rate of production or consumption of charge per unit volume. The vector form of the electric and ionic charge conservation, Eq. (4.75), is given by rUðJÞ 5 S_ c

(4.76)

where the current density is given by Ohm’s law as follows: J 5 σrϕ

(4.77)

where σ is electron or ion conductivity and ϕ is electronic or ionic potential. Combining the current density equation, Eq. (4.77), with the electric and ionic charge conservation, Eq. (4.76), gives rUðσrϕÞ 5 S_ c

(4.78)

The electronic transport form of Eq. (4.78) is   rU σel rϕel 5 S_ c;el

(4.79)

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93

And the ion transport form of Eq. (4.78) is   rU σio rϕio 5 S_ c;io

(4.80)

where the subscript “el” refer to electronic and “io” refer to the ionic. To account for the electron and ion transport through the solid portion of the porous electrodes of the SOFC, an effective diffusion term of the electronic and ionic conductivity is used. One applicable relation of the effective electronic conductivity is as follows: σeff 5

  12ε σ τ

(4.81)

where ε is porosity, τ is tortuosity of the porous electrodes, and σ is the conductivity of the electrode layer material. 4.3.2.1 Electrochemical reaction rate Electrochemical reaction, which takes place at the interface between an electrode and an electrolyte, contributes to the overall reaction rate. The general and elementary form of electrochemical reaction relation is as follows [8]: N X i51

k

ν 0i Mi
!

N X

νvi Mi

(4.82)

i51

where k is a reaction rate constant and M is the molecular weight of different species. According to the law of mass action, which states that the rate of a chemical reaction is proportional to the concentrations of the reacting substances, the rate of reaction for species i, ri, is expressed as   N 0 ri 5 νvi 2 ν 0i k L ðMi Þν i

(4.83)

i51

Furthermore, the rate of electrochemical reaction, ri, is related to the current density, J, by the Faraday’s law as follows: J 5 2 nFri

(4.84)

where n is the number of electrons transferred during the overall half-cell electrochemical reaction in each electrodes and F is the Faraday’s constant. In writing the Faraday’s law, Eq. (4.84), it is assumed that the charge transfer reaction is a single rate-determining step. However, and in reality,

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

the half-cell electrochemical reaction is not a simple one-step reaction. It involves many elementary reactions with the intermediate species. While the knowledge of elementary reactions during the electrochemical reactions in SOFCs is still unknown, it is customary to use the concept of “rate-determining step” of an overall half-cell reaction in the calculation of electrochemical reaction rate. As explained in Chapter 1, Introduction to fuel cells, the Nernst equation is an equation that relates the reduction potential of an half or full cell electrochemical reaction to the concentrations of the chemical species undergoing reduction and oxidation. In another word, the Nernst equation tells us the position of the equilibrium of a reaction. However, the Nernst does not explain about how fast the system may get there. If the reaction proceeds slowly, kinetic limitation, the equilibrium condition may never be observed. For instance, at room temperature and pressure, the diamond reaction to form graphite is thermodynamically unstable. While the vast activation energy for the re-orientation of atoms kinetically limits this reaction such that it is never in practice observed, the diamond is technically at metastable state. The same phenomenon is encountered in many electrochemical situations. Reactions are prevented from proceeding to their equilibrium state by their kinetic limitations. Therefore, many researches aim at overcoming the natural kinetic tardiness of the surface reactions of small, nonpolar molecules such as hydrogen and oxygen in SOFCs. In the absence of viable experimental kinetic data, there are two important expressions, namely the Tafel law and the Butler
Volmer equation that are generally used for modeling of electrochemical reaction in SOFCs. These equations relate the rate of the electrochemical reaction, the current density, to the overpotential and the concentrations of reactant and product. The Tafel law, which is for a single electrode and is for irreversible anodic or cathodic process, relate the overpotential, η, to current density, J, as follows:   J log 5 Aη (4.85) J0 The constant A is the Tafel slope with unit 1/V and is usually close to a half-integer multiple of F/RuT (normally less than or equal to nF/RuT) in which F is Faraday’s constant and Ru is the universal gas constant. J0 is

Fundamentals of electrochemistry

95

the exchange current density, with unit A/m2, and is by definition the current density drawn at zero overpotential during the electrochemical reaction. Obviously, the Tafel law does not apply to a reverse reaction that may occur in SOFC. The Butler
Volmer equation, the most general equation of electrochemical kinetics, approximates the real current density
overpotential relation for a reversible process as follows:      αa Fη 2 αc Fη J 5 J0 exp 2 exp (4.86) Ru T Ru T The Butler
Volmer equation is for all cases, either anodic or cathodic current, that may flow depending on the sign and magnitude of the overpotential. The Butler
Volmer equation is highly adaptable because of the following reasons: • J0 is an empirical quantity. • It agrees with the Nernst equation when current density is zero, J 5 0. Therefore, for a very fast reaction where exchange current density is infinity (J0-N), the Butler
Volmer equation gives the same potential difference as the Nernst equation. This is equally even true under high-resistance conditions. And finally • It agrees with the Tafel equation when either the anodic or cathodic term dominates. Typically for highly irreversible reactions (very low exchange current density, J0) and for large overpotential, significant current density is drawn. In a reversible reaction and for very low overpotential, η (where η is in the order of RuT/F B 25 mV), the exponentials in the Butler
Volmer equation is linearized and take the form of     αa Fη 2 αc Fη ðαa 1 αc Þ exp 2 exp 5 (4.87) Ru T Ru T Ru T Or by using the linearized form (Eq. 4.87), the Bulter
Volmer equation (Eq. 4.86) takes the form of   ðαa 1 αc Þ J 5 J0 η (4.88) Ru T The linearized Butler
Volmer equation is only applicable to electrochemical processes occurring exclusively at low current density, such as

96

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

electroplating or electrochemical impedance spectroscopy (where overpotential is less than about 25 mV at room temperature, RuT/F , 25 mV).

4.3.3 Some applicable boundary conditions for solid oxide fuel cells In order to solve the nonlinear governing equations, the multicomponent mass transfer and charge (electron and ion) transfer, two boundary conditions, one boundary condition for each physics are required. Specifically, the boundary conditions at the catalysts and electrolyte layers relate to the charge (electron or ion) transfer. The mathematical expressions of the mentioned boundary conditions are described in following sections. 4.3.3.1 Inflow boundary conditions The boundary conditions mass and mole fraction, molar concentration, and density at the inlet to the anode and cathode channels are as follows, respectively: • Mass fraction, y, boundary condition is yi 5 y0 •

The mole fraction, x, boundary condition is xi 5 x0

•

(4.90)

The molar concentration, c, boundary condition is ci 5 c0

•

(4.89)

(4.91)

The density, ρ, boundary condition is ρi 5 ρ0

(4.92)

where i subscript stand for species i. The first two boundary conditions, the mass and mole fraction, are the two most common boundary conditions used in modeling the SOFC. Note that the number of boundary conditions at the inlet depend on the total number of species. For instance, if “N” is the total number of species available in the mixture “N 2 1” boundary conditions should be specified. The neutral species quantity is determined by Eq. (4.2) or (4.6). 4.3.3.2 Outflow boundary condition At the outlet of the anode and cathode channels, the diffusion term is neglected. The convective flux boundary condition at the outlet of the

Fundamentals of electrochemistry

97

electrode channel is dominant. So the mathematical form of the outflow boundary condition is originated from negligibility of the diffusion flux at the outlet of the anode and cathode channels. Thus, the outflow boundary condition can be written as follows:   nU 2ρmix Dij ryi 5 0 (4.93) where n is the unity vector perpendicular to the outlet, Dij is the ordinary diffusion coefficient, y is the mass fraction, and ρmix is the density of the mixture. 4.3.3.3 Insulation boundary conditions The mass insulation boundary condition is applied to the electrode walls. The mass flux normal to all surfaces of the electrolyte are zero since the electrolyte is assumed to be impermeable to gases. The insulation boundary condition at the electrode wall is as follows:

DGM nU 2ρmix Di;j;eff ryi 1 ρmix vyi 5 0 (4.94) where DijDGM is the combined effective binary diffusion coefficient based on DGM. The insulation boundary condition for the electronic potential, ϕel, and ionic potential, ϕio, are given by, respectively:   nU 2σel rϕel 5 0 (4.95)   nU 2σio rϕio 5 0

(4.96)

where σ is the conductivity of the electrode layer material. Equations (4.95) and (4.96) are applied to all outer electrode surfaces except the outer layer (which is called electrode current collector) and the interface between electrodes and electrolyte, because they are insulated to electrical and ionic current. Also, the electrical insulation boundary condition, Eq. (4.95), is applied to all surfaces of electrolyte because they are also insulated to electrical current. 4.3.3.4 Electrical potential boundary condition At the outlet side of the SOFC anode, where the current from the anode side is collected, a zero-voltage electronic potential, ϕel, boundary condition is applied as follows:

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

ϕel 5 0

(4.97)

Also, at the cathode current collector which is located at the outer layer of the cathode the electronic potential, ϕel, is equal to the operating cell voltage, Vcell, is applied as follows: ϕel 5 Vcell

(4.98)

4.3.3.5 Axial symmetry boundary condition Due to axial symmetry at the center of the tube in the tubular SOFC, the mass transport boundary condition, zero Neumann boundary condition, or no-flux boundary condition, is applicable. 4.3.3.6 Continuity boundary condition The continuity boundary condition is applicable for all interior boundaries, such as the interior boundaries between anode and cathode channels and the anode and cathode electrode, of SOFC except those for which boundary conditions are mentioned before. The mathematical formulation of the continuity boundary condition for ion and mass transport at mentioned interior boundaries are as follows, respectively: The ion current continuity boundary condition which is maintained at interfaces between the electrodes and electrolyte is given by       nU 2σio rϕio in 5 nU 2σio rϕio out (4.99) The mass transport continuity boundary condition continuity which is maintained at interfaces between channels and electrodes boundaries is given by h

i h

i DGM DGM nU 2ρmix Di;j;eff ryi 1ρmix vyi 5 nU 2ρmix Di;j;eff ryi 1ρmix vyi in

out

(4.100) The continuity also holds for the electrical current through SOFC boundaries.

References [1] W.G. Bessler, S. Gewies, Gas concentration impedance of solid oxide fuel cell anodes II. Channel geometry, J. Electrochem. Soc. 154 (2007) B548
59. [2] Y.A. Cengel, M.A. Boles, Thermodynamics: An Engineering Approach, fifth ed., McGraw-Hill Science, 2005.

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[3] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, John Wiley & Sons, 1960. [4] S. Litster, N. Djilali, Two-phase Transport in Porous Gas Diffusion Electrodes, Institute for Integrated Energy Systems, University of Victoria, Canada, 2004. [5] R. Krishna, J.A. Wesselingh, The Maxwell-Stefan approach to mass transfer, Chem. Eng. Sci. 52 (1997) 861
911. [6] D.A. Nield, A. Bejan, Convection in Porous Media, third ed., Springer, 2006. [7] B.A. Haberman, J.B. Young, Three dimensional simulation of chemically reacting gas flows in the porous support structure of an integrated-planar solid oxide fuel cell, Int. J. Heat Mass. Transf. 47 (2004) 3617
3629. [8] X. Li, Principles of Fuel Cells, Taylor & Francis, New York, 2006.

Further reading E.L. Cussler, Diffusion-Mass Transfer in Fluid Systems, Cambridge University Press, New York, 1997. H. Zhu, R.J. Kee, A general mathematical model for analyzing the performance of fuel-cell membrane-electrode assemblies, J. Power Sources 117 (2003) 61
74.

CHAPTER 5

Fundamental of heat transfer Contents 5.1 Different modes of heat transfer 5.1.1 Conduction heat transfer 5.1.2 Convection heat transfer 5.1.3 Radiation heat transfer 5.2 Energy conservation 5.2.1 Heat equation in electrolytes 5.2.2 Heat equation in porous electrodes 5.2.3 Heat equation in channels 5.3 Solid oxide fuel cell’s source terms 5.3.1 Joule or Ohmic heat source 5.3.2 Irreversible heat source 5.3.3 Reversible heat sources 5.3.4 Heat source generated by chemical reactions 5.4 Some applicable boundary conditions for solid oxide fuel cells 5.4.1 Specified temperature 5.4.2 Thermal insulated boundary 5.4.3 Specified heat flux 5.4.4 Continuity 5.4.5 Outflow 5.4.6 Symmetry 5.4.7 Surface-to-ambient radiation References

103 103 107 109 113 115 116 117 118 118 118 118 119 120 120 120 121 121 121 122 123 124

As stated before, all chemical and electrochemical reactions occurring in fuel cells especially in solid oxide fuel cells (SOFCs) generate heat and significant temperature gradient within the cell. Therefore, it is essential to understand the fundamentals of heat transfer as applied to SOFCs. Example of heat transfer analysis in SOFC is the effect of burning of hydrocarbons such as methane on the water
gas shift and reforming reactions. Heat transfer deals with the rate of heat transfer between physical systems. According to second law of thermodynamics, the heat transfer Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells DOI: https://doi.org/10.1016/B978-0-12-815753-4.00005-1

© 2020 Elsevier Inc. All rights reserved.

101

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

direction is always from the higher temperature medium to lower temperature medium and is normal to the plane of temperature differences. Heat transfer processes set limits to the performance of different SOFC components and systems. Heat transfer processes are classified into three types: conduction, convection, and radiation. Conduction heat transfer is transfer of heat through matter (i.e., solids, liquids, or gases) without bulk motion of the matter. In other words, conduction is the transfer of energy from the more energetic to less energetic particles of a substance due to interaction between them. However, convection heat transfer is due to moving fluid. The fluid is different fuels as well as the air. In convection heat transfer, the heat is moved through bulk transfer of a nonuniform temperature fluid. Finally, radiation heat transfer is energy emitted by matter in the form of photons or electromagnetic waves. Radiation can take place through space without the presence of matter. In fact, radiation heat transfer is highest in a vacuum. Radiation can be important even in situations in which there is an intervening medium. Therefore, in SOFCs where the temperature is high (almost 500°C
1000°C), radiation heat transfer is considerable. The heat generated in the SOFC components is the sum of three heat sources: (1) heat generated from the electrochemical reactions that is applied to the catalyst layers, (2) heat generated due to the cathode and anodic loss arising by the resistance to electron and ion pass, and (3) heat generated due to the Ohmic heating in the electrolyte due to the conduction of the oxygen ions. However, it is worth mentioning that the electrodes’ ionic conductivity is significantly low compared with its electronic conductivity. Therefore, it is reasonable to ignore the Ohmic heating caused by electron that pass within the electrodes. In other words, the heats that are generated within an SOFC are principally due to heat generated from the variation of the water entropy formation, the Ohmic loss, the heat sources due to the activation and the concentration over potentials. The content of this chapter is intended to elaborate in some detail the different modes of heat transfer and their application in different parts of SOFCs. The chapter also provides basic tools to enable the readers to estimate the magnitude of the heat transfer rates for different components and sections of SOFCs components. The chapter presents the general governing equations that deals with different modes of heat transfer and discusses their application in an SOFC. In addition, the chapter introduces different source terms that are related to SOFC and discusses the relative

Fundamental of heat transfer

103

equations. Finally, the chapter introduces all possible boundary conditions which are applicable to any SOFCs.

5.1 Different modes of heat transfer As mentioned, it is of great interest to present the relationship between the chemical reactions, which are temperature dependent and take place in any SOFCs, and the heat transfer phenomena. For instance, when the hydrocarbon fuel and oxidant are fed to the cell, in a single-chamber SOFCs (SCSOFCs) scheme, simultaneously it is certain that the fuel combust and the cell temperature rises. The main objective of this section is to describe the major sources for heat transfer in any SOFCs. The purpose is to determine how the sources/sinks are originated and where they are located in order to model, simulate, and improve the overall cell performance of SOFCs. Heat transfer processes in any SOFCs are as follows: • Conduction heat transfer through solid layers mostly in electrolyte layer and interconnects. • Heat transfer in porous electrode layers. • Convection heat transfer between the bulk motion of the fluid and the solid surface. For instance, transfer of heat between fuel or air flow and wall of anode and cathode electrodes, respectively. • Surface-to-surface radiation. • Heat generation or consumption due to ion and electron transport resistance as well as chemical and electrochemical reactions. Fig. 5.1 schematically depicts the different modes of heat transfer within an SOFC.

5.1.1 Conduction heat transfer Conduction heat transfer is the transfer of heat by means of molecular excitement within a material without bulk motion of the mater. Conduction heat transfer mainly occurs in solids or stationary mediums such as fluids at rest. For instance, transfer of heat in solids is due to the combination of lattice vibrations of the molecules and the energy transport by free electrons, while in gases and liquids it is due to the collisions and diffusion of the molecules. To examine conduction heat transfer let us, for instance, look at the _ (W), through the solid electrolyte layer steady-state heat transfer rate, Q

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Figure 5.1 Different modes of heat transfer occurred within SOFC layers.

thickness, Δx, which is a function of the hot fluid temperature, TH, and cold fluid temperature, TC, the geometry and property are given as _ 5 f ðTH ; TC ; geometry and propertyÞ Q

(5.1)

where the hot fluid, TH, and the cold fluid, TC, temperatures are in absolute Kelvin. It is also possible to express the heat transfer rate based on the hot and cold fluid temperature difference, TH 2 TC, as _ 5 f ½ðTH
TC Þ; geometry and property Q

(5.2)

Fourier’s law of heat conduction relates the heat transfer to mechanical, thermal, and geometrical properties of the medium. Fourier has shown that heat transfer rate is proportional to the temperature difference across the solid layer and the heat transfer area and inversely proportional to the solid layer thickness. That is, Heat transfer rate ~

ðAreaÞðTemperature differenceÞ ðAÞðΔT Þ 5 Thickness Δx

(5.3)

The cross-sectional area, A, is in square meter and the thickness of the slab, Δx, is in meter. The proportionality factor in Eq. (5.3) is replaced by

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105

transport property (k) called thermal conductivity (W/mK) which is a scalar property. Therefore, Eq. (5.3) becomes: _ 5 kA TH 2 TC 5 2 kA TC 2 TH 5 2 kA ΔT Q Δx Δx Δx

(5.4)

Thermal conductivity is the measure of ability of a material to conduct heat. Thermal conductivity is a well-tabulated property for a large number of materials and can be found in the different heat transfer or thermodynamics references. In the limit, the heat transfer rate equation, Eq. (5.4), for any temperature difference, ΔT, across a slab length, Δx, as both approach




_ cond;n 5 2 kA dT Q dx

(5.5)

dT K dx m

is the temperature gradient as shown in Fig. 5.2. The minus sign appearing in the above equation is due to heat transfer and temperature gradient directions are in opposite direction. By rearranging Eq. (5.5) and comparing with electric current flow, the conduction thermal resistance in Cartesian coordinate, Rcond, is as follows: Rcond 5

Figure 5.2 Heat conduction mechanism.

Δx kA

(5.6)

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Conduction thermal resistance, Rcond, is the measure of wall resistance against heat flow. It is obvious that the thermal resistance, Rcond, increases as thickness increases and as surface area and thermal conductivity decreases. Conduction thermal resistance for cylindrical and spherical coordinate is determined from the one-dimensional energy equation in the relative coordinate and are as follows, respectively [1]: Rcond 5

lnðro =rin Þ 2πk

Rcond 5

1 rin

2

4πk

1 ro

(5.7)

(5.8)

where ro and rin are the outside and inside diameters of the cylinder as well as sphere. The general steady one-dimensional conduction heat transfer equation with no generation is written as   1 d N dT k R 50 (5.9) RN dR dR The general unsteady one-dimensional conduction heat transfer equation with source term is written as   1 d @T N dT R (5.10) k 1 qw 5 ρCp RN dR dR @t where R and N in both Eqs. (5.9) and (5.10) are x and 0 for slab, r and 1 for cylinder, and r and 2 for sphere, respectively. q000 (W/m3) is the heat generation, ρ (kg/m3) is density, Cp (kJ/kg  K) is heat capacity, and t (s) is the time. For constant thermos-physical properties Eq. (5.10) becomes   1 d @T N dT R (5.11) k 1 qw 5 ρCp RN dR dR @t
2 where α 5 ρCk p ms is thermal diffusivity. The rate of conduction heat transfer for isotropic medium is a vector quantity. The general three-dimensional constant properties heat conduction equations for isotropic medium in rectangular (x, y, z), cylindrical (r, ϕ, z), and spherical (r, ϕ, θ) coordinates are as follows, respectively:

Fundamental of heat transfer

@2 T @2 T @2 T qw 1 @T 1 1 1 5 @x2 @y2 @z2 k α @t     1@ @T 1 @ @T @2 T qw 1 @T r r 5 1 2 1 2 1 r @r @r r @φ @φ @z k α @t

107

(5.12)

(5.13)

    1 @ 2 @T 1 @2 T 1 @ @T qw 1 @T r sinθ 5 1 2 2 1 2 1 2 2 r @r @r @θ k α @t r sin θ @φ r sin θ @θ (5.14)
W  The rate of conduction heat flux for anisotropic medium, ~ q_ m2 , is also a vector quantity and in Cartesian coordinate system it is as follows @T @T @T @T @T @T 1 kxy 1 kxz Þ^i 2 ðkyx 1 kyy 1 kyz Þ^j @x @y @z @x @y @z @T @T @T 2 ðkzx 1 kzy 1 kzz Þk^ @x @y @z

~ q_ cond 5 2 ðkxx

(5.15) Since the physical properties of all materials used in different layers of SOFCs do not vary with the direction [2], thermal conductivities of these materials are also scalar quantity and therefore Eq. (5.15) is rewritten as @T ^ @T ^ @T ^ ~ q_ cond 5 2 k i2k j2k k 5 2 krT @x @y @z

(5.16)

rT is the temperature gradient and is given as rT 5

@T @T @T ^ ^i 1 ^j 1 k @x @y @z

(5.17)

5.1.2 Convection heat transfer The other mode of heat transfer to be examined is convection heat transfer. Convection heat transfer is the energy transfer between two mediums: typically a surface and fluid that moves over that surface, as shown by Fig. 5.3. In convective heat transfer, heat is transferred by diffusion (conduction) and by bulk fluid motion (advection). Diffusion contribution to convection heat transfer compare to advection is minimal and the advection is the dominant mode. An example of convection heat transfer in an

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Figure 5.3 Convection heat transfer mechanism.

SOFC is the flow of fuel inside anode of the fuel cell, as shown in Fig. 5.3. The goal of convection heat transfer is to determine the flow and temperature behavior of the fluid motion near the surface. When flow passes over a surface, a thin layer of slowly moving fluid, called “boundary layer,” with unknown thickness exists close to the wall. In this region, fluid experiences velocity and temperature differences. A typical example of this phenomena happens at inlet of anode and cathode channels. The fully developed thermal boundary layer forms at the anode and cathode channels inlet when it stops showing any variation in the gas mixture temperature gradient. This is due to the fact that the wall temperature does not vary that much, but the fluid regime experiences large amount of thermal gradients due to the thermal sources that occur within the cell. The boundary layer is not a property and it depends on flow velocity (Reynolds number), structure of the wall surface, pressure gradient, and Mach number. Outside this layer, temperature and velocity are roughly uniform. But most flow regimes in SOFCs system are considered as compressible flow with Mach number smaller than 0.3. _ conv ) from/to the surface to/ The rate of convection heat transfer (Q from the fluid is given by Newton's Law of Cooling as [3] _ conv 5 hAs ðTs 2 Tf Þ Q

(5.18)

The quantity h (W/m2k) is called convective heat transfer coefficient, As is the area, and Ts and Tf are the solid surface and fluid absolute temperature, respectively. For many situations of practical interest, the convective heat transfer coefficient quantity, h, is determined mainly through experiments. The _ conv ) in SOFC is obtained by average convective heat transfer rate (Q

Fundamental of heat transfer

_ conv 5 hAs ðTs 2 Tf Þ Q

109

(5.19)

where the average convective heat transfer coefficient, h, is obtained by integrating Eq. (5.19) over the entire surface and is as follows: ð 1 h5 hdAs (5.20) L As where L is the surface length. From dimensionless boundary layer conservation equations, the local and average convection coefficients for a surface in low-speed, forced convection with no phase change are determined by Nusselt number (Nu) as follows, respectively: x  hx Nux 5 5f ; Re; Pr (5.21) k L Nu 5

hL 5 f ðRe; PrÞ k

(5.22)

Here, k is thermal conductivity of the fluid, Re is Reynolds number (Re 5 uNx/v) and Pr is Prandtl number. For a prescribed geometry, such as flat plate in a parallel flow, under a variety of test conditions (i.e., varying velocity, uN, plate length, L, and fluid nature such as air, water, and oil), there will be many different values of Nusselt number corresponding to a wide range of Reynolds and Prandtl numbers. The results for Nusselt number are presented on a log
log scale in the form of: Nux 5

hx 5 CRem Prn k

(5.23)

C, m, and n are constants and vary with the nature of the surface geometry and type of flow. Just like conductive thermal resistance, a convective thermal resistance (Rconv) is also associated with heat transfer by convection as follows: Rconv 5

1 hA

(5.24)

5.1.3 Radiation heat transfer All bodies at a temperature above absolute zero emit thermal radiation. The origin of thermal radiation, radiation extends between 0.1 and 100 μm, is electromagnetic radiation. Thermal radiation, also called radiation

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

heat transfer, is created by the thermal motion of particles in matter. The high-operating temperature of the SOFC suggests that radiation heat transfer is an important mode of energy transfer and must be included for an accurate model. The nature of radiation heat transfer is by photons or by electromagnetic emissions. Radiation heat transfer is a function of frequency (υ) and wavelength (λ) attributes to radiation. The amount of emitted energy from a surface at a given wavelength depends on the material, condition, and temperature of the body. Thermal radiation includes the entire visible and infrared as well as a portion of ultraviolet radiation. Radiation heat transfer does not require the presence of any matter and is maximized in vacuum. Surfaces that their radiation properties are independent of direction are called diffuse surfaces and if their radiation properties are independent of wavelength they are called gray surfaces. Surfaces that absorb all thermal radiation energy that are incident upon it, regardless of its wavelength and its direction, or emit all thermal radiation energy from its surface in all directions are known as blackbody surfaces. The thermal energy radiated per unit area from a blackbody is given by Joseph Stefan, also known as Stefan
Boltzmann law, as follows [3]: q_ rad;b 5 σT 4

(5.25)

q_ rad;b is the blackbody emissive power, σ is the Stefan
Boltzmann constant and is equal to 5:67 3 1028 m2WUK 4 , and T is the absolute temperature of the blackbody surface. The magnitude of radiation energy per unit area _ rad , is less than the magnitude of a blackbody surfrom any real surfaces, Q face and is defined by [3] _ rad 5 ε_qrad;b 5 εσT 4 Q

(5.26)

where ε is a property and is called the emissivity. Radiation heat transfer also occurs between surfaces inside different sections of SOFC. An example is the radiative heat exchange between the SOFC electrodes outer surfaces and its channels wall. The net rate of radiative heat transfer _ rad;env , and any two surfaces, between a surface and its surrounding, Q _ Qrad;sur , as shown in Fig. 5.4 are given by, respectively [3] 4 _ rad;env 5 εσAs ðTs4 2 Tamb Þ Q

(5.27)

4 4 _ rad;sur 5 εσAs ðTs;1 2 Ts;2 Þ Q

(5.28)

where Ts and Tamb are the surface and ambient absolute temperatures.

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111

Figure 5.4 Radiation heat transfer mechanism.

However, the radiation intensity, through emission of the semitransparent materials, depends on the temperature field and therefore cannot be decoupled from the overall energy equation. In addition, the solution of radiative transfer equation depends on the approximation methods, the boundary conditions, and the radiation properties. In the next two subsections, two methods on the transient temperature distribution solution, namely Schuster
Schwartzchild two-flux approximation and Rosseland approximation, are presented and discussed, respectively. 5.1.3.1 Schuster
Schwartzchild two-flux approximation The radiative heat transfer process involves radiative transfer in electrodes, electrolyte, and participating gases in the channels. Schuster
Schwartzchild two-flux approximation provides a simple one-dimensional solution to the radiative transfer equation for surface-to-surface radiation exchange in the channels, assuming that surfaces are gray, isothermal, non-scattering in the yttria-stabilized zirconia (YSZ) electrolyte, and plane-parallel medium between two black walls at equal temperatures [4]:
 (5.29) qR 5 n2 σ T 4 2 Tw4 e22ðτL 2τÞ where σ is the Stefan
Boltzmann constant, T and Tw are the absolute temperatures of the medium and walls, respectively, and τ is the optical thickness. The radiative heat flux is assumed to be an isotropic function of the propagation direction, but different, over the upper and lower hemisphere. For a medium confined between two isothermal, parallel black plates at temperatures Ttop and Tbottom and separated by a distance L (thickness of the medium(, the two-flux model gives the radiative intensity as [4]:

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

qR 5 C1 e22ζL 1 C2 e22ζx

(5.30)

4 C1 5 2 σðTtop 2 T 4 Þe22ζL

(5.31)

4 2 T 4Þ C2 5 2 σðTbottom

(5.32)

where

and ζ is the absorption coefficient. The radiation transport is coupled with the overall energy conservation by the divergence of the radiative heat flux as a (negative) source term in the energy equation. 5.1.3.2 Rosseland approximation The presence of Nickel doping causes the anode to absorb high amount of thermal radiation, which results in optically thick (τ .. 1) behavior. It is shown that the radiative heat flux qR can be estimated with sufficient accuracy through Rosseland approximation [4]: qR 5 2 krad rT

(5.33)

where krad is the radiative conductivity and defined as krad 5

16n2 σT 3 3β R

(5.34)

where the Rosseland mean absorption coefficient β R can be evaluated from the integration of the spectral absorption coefficient [4] ðN  2  nλ dIbλ n2 dIbλ dλ= dλ (5.35) 5 βR ζ λ dT dT 0 where nλ , Ibλ , and Iλ are the spectral index of refraction, spectral blackbody intensity, and spectral radiative intensity, respectively. Although the Rosseland approximation results in extremely convenient form, it is worth noting that this diffusion approximation is not valid near a boundary and the optically thick assumption should be used with caution. The knowledge of the radiative properties, namely absorption coefficient, ζ, and refractive index, n are needed to predict the radiative heat transfer inside SOFC electrodes and electrolyte. Most SOFCs use yttriastabilized zirconia (YSZ) as the electrolyte, strontium-doped LaMnO3 as the cathode, and nickel-doped YSZ as the anode. For typical operating

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113

temperatures of 900K
1100K and n 5 1.8, over 90% of the emissive power is contained within the near to mid-infrared spectral region, 0:9 μm , λ , 7:8 μm. The absorption coefficient of YSZ electrolyte is obtained from TR 5 e2ζL

(5.36)

where TR is the transitivity, ζ is the absorption coefficient, and L is the thickness of the medium. The multiplication of these two parameters gives the optical thickness τ. However, Mahene et al. [5] reported that the effect of radiative heat transfer within electrode
electrolyte interfaces is small compared to conductive heat transfer. The effect of radiative heat transfer within cathode
electrolyte
anode is respectively small compared to conductive heat transfer. The mean penetration distance for radiative heat transfer in NiYSZ anodes are in the order of nanometers. This means that radiation does not need to be considered in the energy balances within the SOFCs, for an intermediate temperature. If thermal radiation needs to be considered or not depends on several parameters such as temperature differences, materials, gas composition, gas pressure, fuel cell/stack design, and wavelength. With or without radiative heat transfer within the electrolyte interfaces, the effect is negligible on the average cell temperature, (1%
0.9%) temperature difference across the electrolyte at temperatures relevant to SOFCs.

5.2 Energy conservation The equations of heat transfer are derived from the first law of thermodynamics, commonly referred to as the principle of conservation of energy. The conservation statement for energy is written as 0 1 0 1 Rate of change Net rate of energy @ of energy A 5 @ into the cell A 1 convection 1 1by 0 0 within the cell (5.37) Net rate of Net rate of C B work done by C B heat addition C C B B @ by conduction including A 2 @ the cell on A surroundings heat generation The mathematical version of the principle of conservation is as follows:

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

@ @ @ @ ðρeÞ 5 2 ðρueÞ 2 ðρveÞ 2 ðρweÞ @t @x @y @z @qy @qx @qz 2 2 2 1 qw @x @y @z @ðpuÞ @ðpvÞ @ðpwÞ 2 2 2 @x @y @z 1 ρgx 1 ρgy 1 ρgz

(5.38)

@ðuτ xx Þ @ðvτ xy Þ @ðwτ xz Þ 1 1 @x @z @y @ðuτ yx Þ @ðvτ yy Þ @ðwτ yz Þ 1 1 1 @x @y @z @ðuτ zx Þ @ðvτ zy Þ @ðwτ zz Þ 1 1 1 @y @x @z 1

where e is the specific energy, the sum of internal and kinetic energies, ρ is density and the velocity component in x-direction is u, in y-direction v, and in z-direction w. qw is the volumetric heat source term, p is the absolute pressure, and heat conduction in x-direction is qx, in y-direction qy, and in z-direction qz. The acceleration component due to gravity in x-direction is gx, in y-direction gy, and in z-direction gz. τ ij with a double subscript notation is the shear stress tensor. The first subscript for shear stress indicates the shear that acts normal to plane of the substance and the second subscript indicates the direction of stress. In SOFCs, it is assumed that the contributions of the kinetic energy and the gravitational potential energy is negligible. With the mentioned assumption the conduction heat transfer equations (Eqs. 5.16 and 5.39) reduces to one as follows: ρ

@e 1 ρ~ v :re 5 r:ðkrT Þ 1 qw 1 wp 1 qvh @t

(5.39)

According to mass conservation law, the density of the material does not vary with time and thus comes out of the time derivative in Eq. (5.39). The pressure work,wp , in conduction heat transfer equation, Eq. (5.39), is expressed as     T @ρ  @P (5.40) 1 V :rP wp 5 2 P ρ @T  @t

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115

For low-speed flow, the Mach number, V - :rP term, is low and is neglected. The viscous heating term, qvh , is caused by viscous friction within the fluid and is determined by   qvh 5 μ r V 1 ðr V ÞT 2 ð2=3Þðr: V ÞI :r V (5.41) where “I” is unity matrix. In SOFCs, the work done by pressure and viscous stresses are also neglected. Considering these assumptions the conduction heat transfer equation, Eq. (5.39), reduces to   @T 1 V :rT 5 r:ðkrT Þ 1 qw ρcp (5.42) @t The vector form of the conduction heat transfer equation, Eq. (5.40), is as follows: ρcp where

D Dt

DT 5 r:ðkrT Þ 1 qw Dt

(5.43)

is vector operator, also called written as DðÞ @ðÞ 5 1 V :rðÞ Dt @t

(5.44)

5.2.1 Heat equation in electrolytes The electrolytes used in SOFCs are made from ceramics. Ceramics are nonporous and completely dense and thus allow no fluid and species to go through it. The electrolyte-governing energy equation is derived from the heat conduction equation, Eq. (5.43), by setting the velocity components equal to zero. The conduction heat equation for electrolytes, Eq. (5.43), is then as follows: ρc

@T 5 r:ðkrT Þ 1 qw @t

(5.45)

where C is the specific heat of the electrolyte layer and is the same for constant-pressure and constant-volume specific heats in solid ceramic (Cp 5 Cv 5 C) where de 5 CdT

(5.46)

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Due to resistance of oxygen ions, movement causes heat generation in electrolyte layer. This phenomenon is explained in detail in Section 5.3 separately.

5.2.2 Heat equation in porous electrodes As opposed to the electrolytes, the electrodes of an SOFC are porous. The modeling and analysis of heat transfer through a porous medium is complex due to its complicated structures. The modeling of porous medium also depend on the manner in which the solid is interconnected because thermal conductivity of solid phase and the fluid are very different. It is generally assumed that local thermal equilibrium between solid and fluid phases are in place. Therefore, solid temperature, Ts, as well as fluid temperature, Tf, are the same (T 5 Ts 5 Tf). Regarding the porosity of the porous electrodes (ε), the solid portion of porous media in a unit volume is the factor (1 2 ε), because this is the ratio of the volume occupied by solid to the total volume of porous electrodes. The conduction heat transfer equation, Eq. (5.42), for solid and fluid phase of porous electrode where the surface porosity and the porosity are the same are as follows, respectively: @Ts 5 ð1 2 εÞr:ðks rTs Þ 1 ð1 2 εÞqw @t

(5.47)

@Tf 1 ðρcp Þf V :rTf 5 εr:ðkf rTf Þ 1 εqw @t

(5.48)

ð1 2 εÞðρcÞs εðρcp Þf

The indexes “f” and “s” refer to the fluid and the solid phases of the porous electrodes. Since thermal equilibrium between solid and fluid phases of the porous electrodes exist, the conduction heat transfer equation for solid and liquid phase of porous electrode, Eqs. (5.47) and (5.48), are combined as follows: ðρcP Þeff

@T 1 ðρcP Þf V :rT 5 r:ðkeff rT Þ 1 qw @t

(5.49)

where the effective volumetric heat capacity at constant pressure, (ρcp)eff, effective thermal conductivity, keff, and effective source term, qweff , are as follows, respectively [6,7]: ðρcP Þeff 5 εðρcP Þf 1 ð1 2 εÞðρcP Þs

(5.50)

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117

keff 5 εkf 1 ð1 2 εÞks

(5.51)

qweff 5 εqwf 1 ð1 2 εÞqws

(5.52)

The effective thermal conductivity, keff, is influenced by different factors such as fluid concentration, particle-to-particle interactions, and a real contact between particles. Several literatures are available for closed-form solutions of the isotropic effective thermal conductivity [8
16]. If the structure and orientation of the porous medium is such that the heat conduction takes place in series, with all of the heat flux passing through both solid and fluid, then the effective thermal conductivity, keff , is the weighted harmonic means of ks and kf [6,7]: 1 12ε ε 5 1 keff ks kf

(5.53)

In general, the effective conductivity obtained by Eqs. (5.51) and (5.53) will provide upper and lower bounds, respectively, on the actual overall conductivity. However, the results from Eqs. (5.51) and (5.53) are the same if and only if ks 5 kf. If ks and kf are not too different from each other, the weighted geometric mean of ks and kf provides good estimation for the overall conductivity, defined by [6] keff 5 k12ε kεf s

(5.54)

The first term (Eq. 5.15) is obtained by assuming that heat conduction in the solid and fluid phases takes place in parallel so that there is no net heat transfer from one phase to the other. In Eq. (5.49) the source term, qw, is related to reversible and irreversible heat generation, Ohmic heat due to ionic and electronic resistance, heat production or consumption due to exothermic and endothermic chemical reactions. They will be discussed in more detail in Section 5.3.4.

5.2.3 Heat equation in channels The channel is given by @T 1 V :rT Þ 5 r:ðkrT Þ 1 qw ρcP ð @t

(5.55)

In SOFC channels, neither electron nor ion passes and no chemical and electrochemical reactions occur and therefore the heat source qw vanishes.

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

5.3 Solid oxide fuel cell’s source terms Different heat sources that take place in SOFCs are as follows: • Joule or Ohmic heat source: heat sources due to resistance that occurs when electrons or ions passes through electrolyte and porous electrodes • Activation and concentration overpotentials heat source: heat sources due to irreversibility • Reversible heat source • Heat source generated by chemical reactions within the cell

5.3.1 Joule or Ohmic heat source Joule heating, also called Ohmic heating, is the heating that occurs when electric or ionic current passes through a metal. In SOFC Joule heating takes place in the electrolyte and porous electrodes. The joule heating due to electron, qwohm;el , and joule heating due to ion passage, qwohm;ion , are determined as follows, respectively [12]: qwohm;el 5 σel rφel :rφel

(5.56)

qwohm;ion 5 σion rφion :rφion

(5.57)

where σel is electron conductivity, σion is ion conductivity, φel is electron potential, and φion is ion potential.

5.3.2 Irreversible heat source Irreversible heat source, qwirr , is mostly related to activation over potential. It takes place in the porous catalyst layers and is determined by [12] qwirr 5 ηact i

(5.58)

where ηact is activation over potential and the current density for the electrode side, i, is calculated by either Toefel’s, Bulter
Volmer’s or any other experimental equation which is fully explained in Chapter 4, Fundamentals of electrochemistry.

5.3.3 Reversible heat sources The heat is generated by electrochemical reaction, hydrogen reduction and oxygen oxidation that occur in anode and cathode catalyst layer respectively. It is also called reversible heat source and is dominated at the

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119

anode electrode catalyst side. The reversible heat source, qwrev , is obtained by [12]   i qwrev 5 T Δs (5.59) zF where Δs is the entropy change of the H2-oxidation half-reaction at the anode electrode or the O2-reduction half-reaction at the cathode electrode, T is absolute temperature, i is current density, z is the number of electrons that participated in electrochemical reactions, and F is the Faraday’s constant.

5.3.4 Heat source generated by chemical reactions Chemical reaction in SOFCs that uses hydrocarbons fuel occur in both anode and cathode sides. The most important chemical reaction of such an SOFCs are water
gas shift and reforming reactions. Let us consider a two-way chemical reaction as follows: k1



aA 1 bB ’







cC 1 dD

(5.60)

k2

where A and B are the reactants, C and D are the products, a, b, c, and d are the stoichiometric coefficients of the reaction, and k1 and k2 are the reaction rate of forward and backward reactions, respectively. The amount of heat produced during an exothermic reaction or consumed during endothermic reactions, q}react , is determined by the chemical kinematics and is obtained by [13] qvreact 5 Rreact ΔHreact

(5.61)

where ΔHreact is the change of enthalpy between products and Rreact is the reaction rate and is as follows: Rreact 5 k1 cAa cBb 2 k2 cCc cDd

(5.62)

where ci is the concentration of i chemical species. In SCSOFC, where the fuel and oxidant mixture are fed to the cell simultaneously, fuel combustion may take place. In SCSOFC, the combustion reaction is one-way reaction and the amount of reaction rate, k2, is negligible compared with the amount of reaction rate, k1.

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

5.4 Some applicable boundary conditions for solid oxide fuel cells Boundary conditions used to solve the general heat equation (Eq. 5.42) depend on the nature of each problem. Some important boundary conditions that are used in any typical SOFC modeling are listed in the following sections.

5.4.1 Specified temperature In most SOFC modeling, the fuel inlet temperature is specified (i.e., T 5 T0 ) and the rest of the temperatures such as the channel walls temperature are evaluated. Fig. 5.5 depicts a typical example of such boundary condition.

5.4.2 Thermal insulated boundary Usually in absence of channel walls temperature, the thermal insulated boundary condition is assumed. The thermal insulated boundary condition is formulated as nUðkrT Þ 5 0

(5.63)

where n denotes to normal vector on the boundary and k is thermal conductivity of the material. Fig. 5.6 depicts different thermal insulated boundaries that is used in SOFC modeling.

Figure 5.5 Specified temperature boundary condition in SOFCs.

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121

Figure 5.6 Different thermal boundary conditions used in SOFCs modeling.

5.4.3 Specified heat flux Typically, any physical boundaries that are in contact with the fluid flow are known as specified heat flux boundary. At this surface conduction, heat flux is equal to convection heat flux. An example of it in SOFC is the boundary between electrodes outer surfaces and fluid stream in SOFC channels. Specific heat flux boundary is written as nUðkrT Þ 5 hðT 2 Tinf Þ

(5.64)

where h is thermal connectivity and Tinf is the fluid temperature. Fig. 5.7 shows an example of specific heat flux boundary condition between electrode wall and working fluid.

5.4.4 Continuity The continuity boundary is a condition at which the inlet heat flux is equal to the outlet heat flux as follows: nUðk1 rT1 Þ 5 nðk2 rT2 Þ

(5.65)

All SOFC interior boundaries follow the continuity boundary condition. Fig. 5.8 depicts an example of continuity boundary condition between electrode and electrolyte layers.

5.4.5 Outflow The outflow boundary, a condition at which only convection heat transfer occurs at the surface, is a common boundary condition for convection-dominated heat transfer at outlet boundaries. An example of this type of boundary is the SOFC channel outlet where the temperature

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Figure 5.7 Specific heat flux boundary condition between electrode wall and working fluid.

Figure 5.8 Continuity boundary condition between electrode and electrolyte layers.

gradient in the normal direction is zero and there is no radiation. The equation used for outflow boundary is as follows: nUðkrT Þ 5 0

(5.66)

The outflow boundary condition at channel outlet is shown in Fig. 5.9.

5.4.6 Symmetry The symmetry boundary, condition at which no heat flux crosses the boundary, is similar to thermal insulation boundary condition discussed in Section 5.2.2. This type of boundary condition is used when the physical geometry of interest and the expected pattern of the thermal solution are identical. Fig. 5.10 illustrates the physical geometry and its symmetrical view and boundary conditions.

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123

Figure 5.9 The outflow boundary condition at channel outlet.

Figure 5.10 The physical geometry and its symmetrical view and boundary conditions.

5.4.7 Surface-to-ambient radiation The surface-to-ambient radiation boundary condition occurs when heat transfer between the boundary and surroundings is by radiation. An example of this type of boundary condition applies between SOFC outer electrode surfaces and the channel walls exterior that contains high

124

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

temperature. The net inward heat flux stemming from surface-to-ambient radiation is as follows:
 4 nðq1 2 q2 Þ 5 εrad σ0 T 4 2 Tamb (5.67) where ðq1 2 q2 Þ is the net heat flux outlet to the boundary, εrad is the surface emissivity, σ0 is Stefan
Boltzmann constant, and Tamb is the ambient temperature.

References [1] R. Foster, M. Ghassemi, A. Cota, Solar Energy: Renewable Energy and the Environment, CRC Press, Taylor & Francis Group, 2009. [2] G. Kaur, Solid Oxide Fuel Cell Components: Interfacial Compatibility of SOFC Glass Seals, Springer, 2016. [3] Y.A. Cengel, A. Ghajar, Heat and Mass Transfer: Fundamentals and Applications, fifth ed., McGraw-Hill Science, 2014. [4] M.F. Modest, Radiative Heat Transfer, second ed., Elsevier, 2003. [5] H. Mahcen, N. Meddour, D. Bechki, H. Bouguettaia, H.B. Moussa, Radiation phenomenon in electrodes/electrolyte interface of solid oxide fuel cells, J. Energy Procedia 50 (2014) 229
236. [6] A. Bejan, Convection Heat Transfer, fourth ed., John Wiley & Sons, 2013. [7] M. Kaviany, Principles of Heat Transfer in Porous Media, second ed., Springer, 1995. [8] M. Kamvar, M. Ghassemi, M. Rezaei, Effect of catalyst layer configuration on single chamber solid oxide fuel cell performance, J. Energy Appl. Therm. Eng. 100 (2016) 98
104. [9] Y. Mollayi Barzi, M. Ghassemi, M.H. Hamedi, A 2D transient numerical model combining heat/mass transport effects in a tubular solid oxide fuel cell, J. Power Sources 192 (2009) 200
207. [10] Y. Mollayi Barzi, M. Ghassemi, M.H. Hamedi, Numerical analysis of start-up operation of a tubular solid oxide fuel cell, Int. J. Hydrogen Energy 34 (2009) 2015
2025. [11] N. Akhtar, S.P. Decent, D. Loghin, K. Kendall, Mixed-reactant, micro-tubular solid oxide fuel cells: An experimental study, J. Power Sources 193 (2009) 39
48. [12] M. Andersson, J. Yuan, B. Sunden, SOFC modeling considering hydrogen and carbon monoxide as electrochemical reactants, J. Power Sources 232 (2013) 42
54. [13] N. Akhtar, Single-Chamber Solid Oxide Fuel Cells: Modeling and Experiments (PhD thesis), University of Birmingham, 2010. [14] N. Akhtar, S.P. Decent, D. Loghin, K. Kendall, A three dimensional numerical model of a single-chamber solid oxide fuel cell, Int. J. Hydrog. Energy 34 (2009) 8645
8663. [15] M.F. Serincan, U. Pasaogullari, N.M. Sammes, A transient analysis of a microtubular solid oxide fuel cell (SOFC), J. Power Sources 194 (2009) 864
872. [16] M. Andersson, J. Yuan, B. Sunden, SOFC modeling considering electrochemical reactions at the active three phase boundaries, Int. J. Heat Mass Transf. 55 (2012) 773
788.

CHAPTER 6

Fundamentals of fluid flow Contents 6.1 Conservation of mass 6.1.1 Mass sources 6.2 Conservation of linear momentum 6.2.1 The Brinkman equation 6.2.2 The Navier
Stokes equations 6.2.3 Body (volume) force 6.3 Boundary conditions 6.3.1 Inlet boundary condition 6.3.2 Outlet boundary condition 6.3.3 Wall boundary condition 6.3.4 Axial symmetry 6.3.5 Continuity References

126 127 132 132 133 134 134 134 136 137 138 139 139

As explained, fuel cells are systems in which flow of multicomponent species occurs. Therefore, the performance of fuel cell is affected by velocity and pressure distributions of each species. For example, the rate of each chemical reaction entirely depends on partial pressure of each species. Furthermore, the knowledge of stream velocity profile is vital in estimation of fuel consumption. Therefore, it is essential to understand the detail behavior of velocity and pressure distributions inside the fuel cell. The schematic of the physics of the fluid flow inside solid oxide fuel cell (SOFC) components is depicted in Fig. 6.1. In the following sections, the physics of fluid flow in SOFCs, the conservation of mass and the conservation of linear momentum, is presented and discussed in detail. In addition, the general governing equation of fluid flow through porous electrode and the governing equation of free flow of fluid in channel are presented, respectively. In presentation of conservation equations, flow is assumed to be compressible, laminar, and all gaseous species behaves as ideal gases which are nearly valid in most SOFC systems. Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells DOI: https://doi.org/10.1016/B978-0-12-815753-4.00006-3

© 2020 Elsevier Inc. All rights reserved.

125

126

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Figure 6.1 The physics of the fluid flow inside an SOFC component.

6.1 Conservation of mass The law of conservation of mass states that the total mass of any system neither increases nor diminishes, it only changes its form. In a close system where no mass is transferred in/out of the system, the mass of the system remains constant over time. However, in an open system like SOFC where the electrochemical reactions take place in porous catalyst layers, the mass of the mixture is no longer constant and it may be created or destroyed. The general mass conservation equation for SOFC systems is written as follows [1]: @ ðρεÞ 1 rUðρuÞ 5 Qm @t

(6.1)

where ρ is fuel density, Qm is a mass source term, and ε is the porosity which is the fraction of the control volume that is occupied by pores. Value of the porosity can vary from zero for pure solid regions such as electrolyte part in the conventional SOFC to unity for domains of free flow which is valid for fluid flow in SOFC channels. Due to high-temperature gradient in SOFC components, the fuel is modeled as an ideal gas and the compressibility effect is determined by [2] ρ5

pMmix RT

(6.2)

where R is the universal gas constant and is equal to 8.314 kJ/kmol  K, p is the absolute pressure and is obtained from the conservation of the linear momentum described in Section 6.2, and T is the absolute temperature.

Fundamentals of fluid flow

127

The molecular weight of the gas mixture, Mmix, depends on mole fraction of each gas species and is determined by [2]: X xi Mi (6.3) Mmix 5 i

where xi and Mi are the mole fraction and the molecular weight of the ith species respectively. For incompressible flow cases, the continuity equation (Eq. 6.1) becomes rUðρuÞ 5 Qm

(6.4)

Eq. (6.4) comes from this fact that for incompressible flow the density stays constant for any particle of flow, which can be expressed as [3] @ ðρεÞ 1 uUrρ 5 0 @t

(6.5)

For steady-state operation of SOFC, the continuity equation becomes [3] rUðρuÞ 5 Qm

(6.6)

6.1.1 Mass sources In the following section, the factors that create mass sources/sink within the SOFC systems are presented. Specifically, the mass source/sink caused by chemical and electrochemical reactions due to their importance in SOFC is given more attention. Note that in SOFC the electrochemical and chemical reactions take place only in the catalyst layers and therefore the mass source/sink term for the production/consumption of gas molecules is zero for gas diffusion electrodes. The reason for introducing the source/sink term (given in Ref. [4]) is that sometimes the reactive layers can extend into the electrodes. 6.1.1.1 Mass sources caused by chemical reactions At the anode side, the production of hydrogen required for electrochemical reactions is done by water
gas shift and reforming chemical reaction for all fuels except hydrogen. In single-chamber SOFC where the fuel and oxidant are fed into the cell simultaneously, the fuel combustion takes place at both sides of the electrodes. However, it is generally assumed that all chemical reactions are taken place at catalyst layers. The effects of these chemical reactions are known as source term for products and sink term for reactant in governing equations.

128

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

To better understand the chemical reactions, methane gas is supplied as fuel into a conventional SOFC where the reforming reactions take place within the anode. Methane is reformed with steam, the so-called methane steam reforming reaction (MSR) is as follows: CH4 1 H2 O23H2 1 CO

(6.i)

Carbon monoxide reacts with water, the so-called water
gas shift reaction (WGSR) is as follows [5]. CO1H2 O2H2 1CO2

(6.ii)

The detail about the reaction mechanisms can be found in Janardhanan and Deutschmann study [6]. When MSR has a significantly higher reaction rate, methane reacts directly with carbon dioxide to produce hydrogen and carbon monoxide, so-called dry reforming, because there is no need of water, and it is given as follows [5]: CH4 1CO2 22H2 1 2CO

(6.iii)

In the above equations, the mass source/sink terms for each species are listed in Table 6.1. In Table 6.1, RSR, RDR, and RSH are reaction rate of steam reforming, dry reforming, and water
gas shift reactions, respectively. As mentioned before, all source/sink terms take place at anode side and there is no source term for the cathode electrode of conventional SOFC. For single-chamber SOFCs, the source/sinks terms given in Table 6.1 is modified. For example, the methane fuel full combustion is as follows [7]: CH4 12O2 22H2 O1CO2

(6.iv)

Table 6.1 The sample source/sink terms for each species taken apart in chemical reactions for a special case in which methane is used as fuel in SOFC.

Species

Source/sink terms of ith species, Ri

Hydrogen, H2 Steam water, H2O Methane, CH4 Carbon monoxide, CO Carbon dioxide, CO2

ð3RSR 1 2RDR 1 RSH ÞMH2 ð2 RSR 2 RSH ÞMH2 O ð2 RSR 2 RDR ÞMCH4 ðRSR 1 2RDR 2 RSH ÞMCO ð2 RDR 1 RSH ÞMCO2

Fundamentals of fluid flow

129

The source/sink terms for the single-chamber SOFC using methane as fuel at anode and cathode side are listed in Tables 6.2 and 6.3, respectively. In Tables 6.2 and 6.3, RFOX is reaction rate of methane full oxidation. 6.1.1.2 Mass sources caused by electrochemical reactions As explained in Chapter 2, Classification of SOFCs, two common electrochemical reactions occur within SOFCs, the oxygen reduction at the cathode catalyst layer and the hydrogen oxidation at anode catalyst layer, as follows, respectively: O2 14e2 -2O22

(6.v)

H2 1 O22 -H2 O 1 2e2

(6.vi)

As seen in Eq. (6.vi), water is also produced at the anode catalyst layer. The rate of consumption (production) of each species taken apart in electrochemical reactions can be calculated by   Ri 5 6 AaðcÞ iaðcÞ Mi =nF (6.7) where Aa and Ac are electrochemically active surface area of the medium per unit volume for anode and cathode catalyst layer, respectively. The values of Aa and Ac depend on the microstructure of catalyst layers and are generally extracted from experiments. ia and ic are anodic and cathodic current densities, respectively, and are obtained from the well-known Bulter
Volmer formula. Mi is the molar mass of ith species and n is the number of electrons taken apart in electrochemical reactions. Number of electrons n 5 4 for oxygen reduction and n 5 2 for hydrogen oxidation. Finally, F is Faraday’s constant and is equal to 96485 C/mol. Table 6.2 The sample source (sink) terms for each species taken apart in chemical reactions for a special case in which methane used as fuel in single-chamber SOFC at anode side.

Species

Source (sink) terms of ith species, Ri

Hydrogen, H2 Steam water, H2O Methane, CH4 Carbon monoxide, CO Carbon dioxide, CO2 Oxygen, O2

ð3RSR 1 2RDR 1 RSH ÞMH2 ð2 RSR 2 RSH 1 2RFOX ÞMH2 O ð2 RSR 2 RDR 2 RFOX ÞMCH4 ðRSR 1 2RDR 2 RSH ÞMCO ð2 RDR 1 RSH 1 RFOX ÞMCO2 ð2 2RFOX ÞMO2

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Table 6.3 The sample source (sink) terms for each species taken apart in chemical reactions for a special case in which methane used as fuel at cathode side.

Species

Source (sink) terms of ith species, Ri

Hydrogen, H2 Steam water, H2O Methane, CH4 Carbon monoxide, CO Carbon dioxide, CO2

0 ð2RFOX ÞMH2 O ð2 RFOX ÞMCH4 0 ðRFOX ÞMCO2

In real operation of SOFCs, the oxidation of carbon monoxide takes place at the anode side and is ignored in most numerical studies. However, for more accurate results, this extra electrochemical reaction should be considered. The oxidation of carbon monoxide is given as [5] CO1O22 -CO2 12e2

(6.vii)

A model that is used for hydrogen and carbon monoxide as electrochemical reactant is given by Andersson et al. as follows [5]:   pffiffiffiffiffiffiffi 0:25 3 1010 exp 2 130;000 :Ru T pO2 Ru T ic 5 Ac : (6.8)  F  

2 2Fηact;c 2Fηact;c 3 exp 2 exp Ru T Ru T   2:1 3 1011 exp 2 120;000 :Ru T Up0:266 H2 O Ru T ia;H2 5 Aa :  0:266 F: Keq;H2 :pH2    

2Fηact;a;H2 2 Fηact;a;H2 3 exp 2 exp Ru T Ru T   0:84 3 1011 exp 2 120;000 URu T Up0:266 CO2 Ru T ia;CO 5 Aa U  0:266 FU Keq;CO UpCO    

2Fηact;a;CO 2 Fηact;a;CO 3 exp 2 exp Ru T Ru T

(6.9)

(6.10)

Keq;H2 and Keq;CO are temperature-dependent equilibrium constant for reactions H2 1 1=2O2 -H2 O and CO 1 1=2O2 -CO2 , respectively. In addition, ηact,c is the activation polarization due to electrochemical

Fundamentals of fluid flow

131

reduction of oxygen at cathode side, ηact,a,H2 is the activation polarization due to electrochemical oxidation of hydrogen at anode side, and ηact,a,CO is the activation polarization due to electrochemical oxidation of carbon monoxide at anode side and are calculated by [5,7,8] ηact;a;H2 5 ϕio 2 ϕel

(6.11)

Nernst Nernst 2 EH ηact;a;CO 5 ηact;a;H2 1 ECO=O 2 2 =O2

(6.12)

ηact;a;H2 5 ϕio 2 ϕel 2 VOC

(6.13)

VOC is open circuit voltage, the Nernst voltage, and is calculated by [9]   247340 2 54:85T Ru T pH2 Ru T   1 VOC 5 (6.14) 1 ln ln pO2 2F 2F 4F pH2 O The Nernst potential, ENerst , is determined by the difference in thermodynamic potentials of the electrode reactions. The Nernst potential of hydrogen-steam mixture as fuel is calculated from Eq. (6.15) [10,11] and for carbon monoxide as fuel is calculated from Eq. (6.16) [12]. The Nernst potential for hydrogen and carbon monoxide is developed for simplified pure mixtures [5]: ! R T p u H2 O Nernst ln (6.15) EH 5 1:253 2 2:4516 3 1024 T 2 pffiffiffiffiffiffiffi 2 =O2 2F pH2 pO2 ! R T p u CO2 Nernst ECO=O 5 1:46713 2 4:527 3 1024 T 2 ln pffiffiffiffiffiffiffi 2 2F pCO pO2

(6.16)

In the above equations neither the local partial pressure of hydrogen, water, or methane influences the Nernst potential for the reaction with carbon monoxide as electrochemical reactant nor the partial pressure of carbon monoxide, carbon dioxide, or methane influences the Nernst potential for the reaction with hydrogen as electrochemical reactant. Finally, the total mass source term, Qm, is the sum of all chemical and electrochemical mass sources/sinks of the species and is as follows: X Qm 5 Ri (6.17) i

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

6.2 Conservation of linear momentum Conservation of linear momentum states that the momentum of any system/body in motion stays constant as long as no external force is exerted to the system. The governing equations for free fluid flow in SOFC channel are the Navier
Stokes equations. In addition, the Brinkman equation is the governing equation for the porous media fluid flow in the porous electrodes.

6.2.1 The Brinkman equation The analysis of shear flow through porous electrodes of SOFC is determined by the Brinkman equation. To account for the viscous transport in the momentum equation, the Darcy’s law is extended by the Brinkman model, while it treats both the pressure and the flow velocity vector as independent variables. In porous media, the flow variables and fluid properties are defined at any point inside the medium by means of averaging the actual variables and properties over a certain volume surrounding the point. This control volume must be small compared to the typical macroscopic dimensions of the problem, but it must be large enough to contain many pores and solid matrix elements. The flow velocity is defined as a superficial volume average, and it corresponds to a unit volume of the medium, including both the pores and the matrix. It is sometimes called the Darcy velocity, defined as the volume flow rate per unit cross section of the medium. Such a definition makes the velocity field continuous across the boundaries between porous regions and regions of free flow. The dependent variables in the Brinkman equations are the Darcy velocity and the pressure. The flow in porous media is governed by a combination of the continuity equation (Eq. 6.1) and the momentum equation, which together form the Brinkman equations as follows [1]:      2μ ρ @u u μ T 1 ðuUrÞ ru 1 ðruÞ 2 ðrUuÞI 5 r: 2pI 1 ε @t ε ε 3ε   Qm 2 μκ21 1 2 u 1 F ε (6.18)

Fundamentals of fluid flow

133

where μ is dynamic viscosity of the fluid, u is the velocity vector, ρ is the density of the fluid, p is the absolute pressure, ε is the porosity, κ is the permeability of the porous electrodes, and Qm is a mass source/sink. The force term, F, accounts for the influence of gravity and other volume forces. Nevertheless, the gravity has minor effect on the SOFC flow regime and is generally By applying the Stokes
Brinkman’s assumption, the term  ignored.  ðuUrÞ u=ε on the left-hand side of Eq. (6.18) vanishes for porous electrodes. Similar to continuity equation, it is common to use ideal gas assumption (Eq. 6.2) to consider compressibility effects on Brinkman equations. Also, for incompressible flow, the density stays constant in any fluid particle and Eq. (6.5) is valid and continuity equation reduces to Eq. (6.6). For steadystate operation of SOFC, the fluid velocity does not change with the time and thus the term @u=@t on the left-hand side of Eq. (6.18) vanishes. The dynamic viscosity for the fluid mixture is determined by [13] X μ5 xj μj (6.19) j

in which xj and μj are the mole fraction and the dynamic viscosity of jth species, respectively. The mole fraction of jth species is obtained from mass transfer physic. Todd et al. [13] reported the methods for the calculation of the thermodynamic and transport properties such as dynamic viscosity of each mixture components used in SOFCs over the temperature of 273K
1473K at ambient pressure.

6.2.2 The Navier
Stokes equations The analysis of flow in the free channel section of SOFCs is governed by a combination of the continuity equation and the momentum equation, which together forms the Navier
Stokes equations. In the free media gas chamber, porosity (ε) is equal to one and permeability (κ) is equal to infinity. Substituting the value of porosity and permeability, Eq. (6.18) reduces to     2μ @u T ρ 1 ρðuUrÞu 5 rU 2pI 1 μ ru 1 ðruÞ 2 ðrUuÞI 1 ρg 1 F @t 3 (6.19) F is the body (volume) force and is discussed in more detail in Section 6.2.3. For incompressible flow, the term 2μ 3 ðrUuÞI is omitted from the left-hand side of Eq. (6.19). Also in stationary operation of SOFC the term ρ @u @t is equal to zero.

134

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

6.2.3 Body (volume) force The body (volume) force term in Brinkman equations is caused by the chemical reactions that take place within porous catalyst layers of the SOFC. The amount of the body (volume) force depend on sum of each species reaction rate tabulated in Table 6.1 for conventional SOFCs and Tables 6.2 and 6.3 for single-chamber SOFCs. The body force is mathematically formulated by [14] 2 3 u X F5 Rj 4 v 5 (6.20) j w in which u, v, and w are velocity vector components. For pure hydrogen as fuel, the body force vanishes due to lack of chemical reactions that occurs in the SOFCs.

6.3 Boundary conditions The proper solution to the governing equations (i.e. mass and momentum equations) needs proper boundary conditions. In this section, the most applicable boundary conditions used in SOFC modeling are presented.

6.3.1 Inlet boundary condition Fig. 6.2 depicts the inlet boundary condition for a planar SOFC at anode and cathode channels. At the anode and cathode channel, the inflow velocity is usually specified as normal and is mathematically formulated as u 5 2 U0 n

(6.21)

where u is velocity vector, U0 is the velocity magnitude of fluid, and n is the boundary normal pointing out of the channel. The minus sign in Eq. (6.21) indicates that the velocity vector is in opposite direction to normal direction “n.” Another inlet boundary condition is the known flow rate. The flow rate is used to calculate the amount of gas or volume of gas that passes through a given section in a unit time. Flow rate is commonly given as standard cubic centimeter per minute (SCCM), a flow measurement term (cm3 /min) at standard conditions (i.e., temperature and pressure or density of the fluid). The standard flow rate sets a standard volumetric flow rate.

135

Fundamentals of fluid flow

Figure 6.2 The schematic of inlet boundary condition of a planar SOFC channels.

The flow occurs across the whole boundary in the direction of the boundary normal and is computed by a surface (3D) or line (2D) integral. The tangential flow velocity is set to zero. Thus, the inlet boundary condition is determined by ð ρ 2 ðuUnÞdA 5 QSCCM (6.22) A ρst where QSCCM is the number of SCCM units and ρst is the standard density. The standard density is determined directly from ρst 5

Mmix Vm

(6.23)

Mmix is mixture molar mass and Vm is standard molar volume which is equal to 0.0224136 m3/mol. The standard density is defined by specifying a standard pressure and temperature, in which case the ideal gas law is assumed: ρst 5

Pst Mmix Ru Tst

(6.24)

Pst and Tst are the absolute pressure and temperature at standard condition (Pst 5 101,325 Pa and Tst 5 273.15K), respectively.

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

6.3.2 Outlet boundary condition Fig. 6.3 schematically depicts the outlet boundary condition at the SOFC channel exit. At the channel’s exit, the normal stress is determined by the pressure. In most cases, the value of the normal stress is approximately equal to the pressure. In addition, the tangential stress component is set to zero. The boundary condition at the channel’s exit is mathematically formulated by    2pI 1 μ ru 1 ðruÞT n 5 2 p0 n (6.25) where p0 is the total pressure at the outlet. When the outlet pressure of the channel (p0) is greater than the pressure on the boundary surface (^p0 ) the backflow is suppressed and the fluid is prevented from entering the channel boundary at the exit. Thus, the following equation is governed at the exit boundary:    2pI 1 μ ru 1 ðruÞT n 5 2 p^ 0 n; p0 $ p^ 0 (6.26) In some cases the normal flow condition, no tangential velocity, is governed at the channel outlet and is formulated as u:t 5 0

(6.27)

where t is the unit tangent vector.

Figure 6.3 The schematic of the outlet boundary condition of a planar SOFC channel.

Fundamentals of fluid flow

137

The laminar outflow condition is typically assumed for the outlet velocity profile. This type of boundary condition is not a very strong assumption for unidirectional flow that is perpendicular to the boundary (see Fig. 6.4). As shown in the Fig. 6.4, a fictitious domain of length Lexit is attached to the channel outlet and the flow is assumed to fully developed laminar flow. This boundary condition is formulated by    Lexit rt U 2pI 1 μ rt u 1 ðrt uÞT n 5 2 pexit n (6.28) where the index “t” stands for tangential direction and pexit is the exit pressure. In case the average outlet velocity or outlet volume flow is specified instead of the pressure, an ordinary differential equation is defined and utilized to calculate pexit in such a way that the desired outlet velocity or volume flow is obtained.

6.3.3 Wall boundary condition Fig. 6.5 depicts the 2-D model of the SOFC wall boundary conditions. For stationary solid wall the no-slip boundary condition, no fluid movement at the wall, is assumed and mathematically formulated as follows:

Figure 6.4 An example of the physical situation simulated when using the laminar outflow boundary condition at SOFC channel outlet. The dashed domain is a fictitious domain.

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Figure 6.5 2-D model of SOFC wall boundary conditions.

u50

(6.29)

where u is velocity vector. However, at the porous electrodes’ exterior wall, the slip boundary condition is used. The slip boundary condition implicitly assumes that there are no viscous effects, hence no boundary layer and no penetration at the wall. It is mathematically formulated as uUn 5 0

(6.30)

where n is the unity normal vector to the wall surface. From the modeling point of view the no-slip boundary condition is a reasonable approximation if the main goal of the wall is to prevent the fluid from leaving the domain and thus

where K is

K 2 ðKUnÞn 5 0

(6.31)

   K 5 μ ru 1 ðruÞT n

(6.32)

6.3.4 Axial symmetry In tubular type due to symmetry (Fig. 6.6), the axial symmetry boundary condition is assumed fluid flow simulation. The symmetry boundary condition implies that no penetration exists at the symmetrical location and the shear stresses vanish as well. This boundary condition is a combination of a Dirichlet and Neumann type of boundary condition for the compressible and incompressible fluid flow cases as follows, respectively:      2 T uUn 5 0; 2pI 1 μ ru 1 ðruÞ 2 μðrUuÞI n 5 0 (6.33) 3

Fundamentals of fluid flow

139

Figure 6.6 The axial symmetry boundary condition in a tubular SOFC.

uUn 5 0;



  2pI 1 μ ru 1 ðruÞT n 5 0

(6.34)

The Dirichlet condition takes precedence over the Neumann condition, and the above equations for the compressible and incompressible fluid becomes uUn 5 0;

K 2 ðKUnÞn 5 0

(6.35)

K is defined in Eq. (6.32).

6.3.5 Continuity The continuity boundary condition is applied for all interior walls between the components except the interior wall of the electrolyte layer. This condition describes the flow field as continuous across the interior boundary.

References [1] D.A. Nield, A. Bejan, Convection in Porous Media, third ed., Springer, 2006. [2] Y.A. Cengel, M.A. Boles, Thermodynamics: An Engineering Approach, eighth ed., McGraw-Hill, 2015. [3] G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 2000.

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

[4] J.C.R. Morales, J.C. Vazquez, D.M. Lopez, J.T.S. Irvine, P. Nunez, Improvement of the electrochemical properties of novel solid oxide fuel cell anodes LSCM-O and LSTMG-O, using Cu
YSZ-based cermets, Electrochim Acta 52 (2007) 7217
7225. [5] M. Andersson, J. Yuan, B. Sunden, SOFC modeling considering hydrogen and carbon monoxide as electrochemical reactants, J. Power Sources 232 (2013) 42
54. [6] V. Janardhanan, O. Deutschmann, CFD analysis of a solid oxide fuel cell with internal reforming: Coupled interactions of transport, heterogeneous catalysis and electrochemical processes, J. Power Sources 162 (2006) 1192
1202. [7] N. Akhtar, S.P. Decent, K. Kendall, Numerical modeling of methane-powered micro-tubular, single-chamber solid oxide fuel cells: an experimental study, J. Power Sources 195 (2010) 7796
7807. [8] R. Suwanwarangkul, E. Croiset, E. Entchev, S. Charojrochkul, M.D. Pritzker, M. W. Fowler, et al., Experimental and modeling study of solid oxide fuel cell operating with syngas flow, J. Power Sources 161 (2006) 308
322. [9] D.H. Jeon, A comprehensive CFD model of anode-supported solid oxide fuel cells, Electrochemica Acta 54 (2009) 2727
2736. [10] Fuel Cell Handbook, seventh ed., U.S. DoE, Morgantown, West Virginia, 2004. [11] W. Winkler, P. Nehter, Modeling solid oxide fuel cell, chapter 2: Thermodynamics of fuel cells, Fuel Cells Hydrogen Energy 1 (2008) 15
50. [12] M. Ni, Modeling of SOFC running pre-reformed gas mixture, Int. J. Hydrogen Energy 37 (2012) 1731
1745. [13] B. Todd, J.B. Young, Thermodynamic and transport properties of gases for use in solid oxide fuel cell modeling, J. Power Sources 110 (2002) 186
200. [14] M.F. Serincan, U. Pasaogullari, N.M. Sammes, Computational thermal-fluid analysis of a microtubular solid oxide fuel cell, J. Electrochem. Soc. 155 (2008) B1117
B1127.

CHAPTER 7

Case studies Contents 7.1 Case study 1: Stationary performance analysis of a dual chamber solid oxide fuel cell with hydrogen fuel 7.2 Case 2: Transient performance analysis of a dual chamber solid oxide fuel cell with hydrogen fuel 7.3 Case study 3: The effect of coplanar and perpendicular catalyst layer configurations on the performance of a single-chamber SOFC References

142 153 160 171

Fuel cells are a brand-new technology for generating energy, which could produce efficient electrical power through direct mixing fuel and oxidant without harming our environment [1]. Nowadays, the solid oxide fuel cells (SOFCs) gain considerable attentions due to their high efficiency and the variety of selecting fuels [2 6]. It is known that SOFCs work in hightemperature ranges which are between 700°C and 1000°C [7]. These highworking temperatures have both positive and negative effects on SOFC commercialization. On one hand it enables direct internal reforming in the anode as well as fuel flexibility, and on the other hand it can lead to crack growth within SOFC components due to thermal stresses [8]. There is an extreme necessity in studying SOFCs in case of having the better management of transferring energy. The studies that have investigated SOFCs behavior can be divided into two categories, namely “numerical” and “experimental” approaches. The experimental studies could be more accurate and their error originates from the measurement tools accuracy which can approach to zero by accurate calibration methods. However, experimental tests are usually expensive, time, and energy consumptive and also are unable to provide details, including the concentration of chemical species, the velocity of fluid, the current density, and so on. Access to these data plays a key role in improving SOFC performance and facilitates its commercialization. Accordingly, the development of numerical studies in this area that provides the details is very useful especially when there has been too little experimental Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells DOI: https://doi.org/10.1016/B978-0-12-815753-4.00007-5

© 2020 Elsevier Inc. All rights reserved.

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information on the transient performances of SOFCs in the open literature [9]. Indeed, a number of studies on SOFC transient and stationary have been done and published in the open literature using numerical approaches [10 21]. Among these studies, Hussein et al. [12] used a two-dimensional model to analyze a planar SOFC behavior under steady-state conditions. They claimed that their model is fuel flexible. The significant note in their study was the consideration of reaction zone layer as finite volume which is closer to reality. However, their model was isothermal and did not account convective terms caused by fluid flow regime. Also, the transient response of SOFC performance was not included in their study. In other study, M. Serincan et al. [13] developed a two-dimensional, axisymmetric transient computational fluid dynamics (CFD) model for an intermediate temperature micro-tubular SOFC. The current density response of the SOFC as a result of step changes in voltage was investigated in their study. However, their model was pure transient and did not provide the prediction of the stationary behavior of the cell. Moreover, the step changes in other input parameters such as inlet temperature and velocity as well as inlet fuel molar fraction were not involved in their study. Recently, X. Ho [9] presented an analysis of transient behavior of an anode-supported SOFC which had been built for steady-state operation performed in his previous study. Step changes of working voltages and fuel composition were applied to the cell, but the step change in inlet temperature and velocity were not involved in his work. In most research works, the electrochemical reactions are assumed to occur in 20 50 µm depth from electrode electrolyte interface which is named reaction zone layer. This concept has been employed by several authors such as Jeon [18], Pasaogullari and Wang [19], Nam and Jeon [20], and Zhu and Kee [21], among others, for steady-state operation of SOFCs. However, to our knowledge, just Ho’s [9] work uses it for transient modeling of SOFCs. The aim of this chapter is to present some case studies predicting stationary and transient performance of single- and dual-chamber SOFCs. Both hydrogen and methane involve in these case studies as fuel. Details of solution procedure are presented for each case.

7.1 Case study 1: Stationary performance analysis of a dual chamber solid oxide fuel cell with hydrogen fuel A single, coflow, dual chamber solid oxide fuel cell is considered as shown in Fig. 7.1A. Fig. 7.1B depicts the schematic of the problem. As shown,

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Figure 7.1 (A) The schematic of a single coflow planar SOFC. (B) The computational domain.

hydrogen as fuel with composition of 95% (mole fractions) is fed to the cell through anode gas chamber and on the other side, dry air as oxidant is fed to the cell through cathode gas chamber. In order to decrease the solution time, the cross-section of the cell which shown as “A” in Fig. 7.1A is considered as the computational domain of current model. As shown in Fig. 7.11B, the computational domain consists of seven layers: (1) anode gas chamber, (2) anode backing layer made of Ni, (3) anode reaction zone layer made of Ni 1 YSZ (yttria-stabilized zirconia), (4) electrolyte layer made of YSZ, (5) cathode reaction zone layer made of LSM 1 YSZ, (6) cathode backing layer made of LSM (lanthanum strontium manganite), and (7) cathode gas chamber. Table 7.1 shows the geometrical parameters of the model. The fluid is assumed to behave as an ideal gas and the flow is twodimensional, compressible, and laminar. All fluid properties are considered temperature dependent. Table 7.2 tabulated thermo-fluid parameters used in this case study as a function of temperature. In addition, the ohmic resistance due to electron transport and thermal diffusion is neglected. This is due to the electronic conductivity of the

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Table 7.1 Geometrical data for case 1. Description

Symbol

Value

Channel width Cell height Anode thickness Anode catalyst layer thickness Electrolyte thickness Cathode thickness Cathode catalyst layer thickness

Wch Hch ta tac te tc tcc

0.5 (mm) 10 (mm) 1000 (µm) 20 (µm) 10 (mm) 50 (µm) 20 (µm)

Table 7.2 Temperature-dependent thermo-fluid parameters used in case study 1 [12,22 26]. Description

Symbol Value

Dimensions

Dynamic viscosity of hydrogen Dynamic viscosity of oxygen Dynamic viscosity of nitrogen Dynamic viscosity of water Thermal conductivity of hydrogen Thermal conductivity of oxygen Thermal conductivity of nitrogen Thermal conductivity of water Specific heat of hydrogen Specific heat of oxygen Specific heat of nitrogen Specific heat of water

μH2

6:162 3 1026 1 1:145 3 1028 3 T Pa
s

μO2

1:668 3 1025 1 3:168 3 1028 T

Pa
s

μ N2

1:435 3 1025 1 2:642 3 1028 T

Pa
s

μH2 O

4:567 3 1026 1 2:209 3 1028 3 T Pa
s

kH 2

0:08525 1 2:964 3 1024 T

W/m
K

kO2

0:01569 1 5:69 3 1025 T

W/m
K

k N2

0:01258 1 5:444 3 1025 T

W/m
K

kH 2 O

20:0143 1 9:782 3 1025 T

W/m
K

cp;H2

13960 1 0:95T

J/kg
K

cp;O2 cp;N2

876:80 1 0:217T 935:6 1 0:232T

J/kg
K J/kg
K

cp;H2 O

1639:2 1 0:641T

J/kg
K

electrodes that is noticeably higher than ionic conductivity. The inertia term in porous electrodes is also neglected due to Stokes Brinkman’s assumption. Table 7.3 tabulated the other input parameters. Note that in order to have more realistic prediction of SOFC behavior, most input parameters in this case study are chosen from Rogers’ experimental test [27].

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Table 7.3 Other input parameters used in case study 1 [12,22 26]. Description

Symbol

Value

Dimensions

Inlet velocity Inlet temperature Total pressure Anode and cathode permeability Universal gas constant Anode thermal conductivity Cathode thermal conductivity Electrolyte thermal conductivity Anode-specific heat Cathode-specific heat Electrolyte-specific heat Anode density Cathode density Electrolyte density

uin T0 P0 κ Ru ka kc ke Cp,a Cp,c Cp,e ρa ρc ρe

0.5 1073 1 10213 8.314 3 3 2 595 573 606 6870 6570 5900

m/s K atm m2 J/mol
K W/m
K W/m
K W/m
K J/kg
K J/kg
K J/kg
K kg/m3 kg/m3 kg/m3

To fulfill the modeling process, it is necessary to determine the boundary conditions for each physics governed in SOFCs. The most common and applicable boundary conditions for each physics are listed and discussed in Chapters 4 6. However, the used boundary conditions for this case are proposed. At the inlet of the anode and cathode gas chamber, the inlet velocity, temperature, and species molar fraction are specified. The inlet velocity and temperature are considered as 5 m/s and 1073K, respectively, and the hydrogen and oxygen inflow mole fraction are 0.95 and 0.21, respectively. It is worth mentioning that water species at anode side and nitrogen species at cathode side are determined to compute from mass constraint (sum of the total mole fractions of the different species existed in a mixture should be equal to unity). At the outlet of the anode and cathode gas chambers, the pressure is equal to atmosphere pressure and the flow is forced to exit the chamber perpendicularly to the outlet. This is possible when a no tangential stress condition changes to no tangential velocity condition. Also, the outlet pressure is adjusted in order to prevent fluid from entering the domain through the boundary. The conduction heat transfer compared to convection heat transfer is negligible at outlet. Similarly, it is assumed that convection is the dominating effect driving the mass flow through the outflow boundary. No-slip boundary condition is applied for chambers walls and electrolyte exterior boundaries since electrolyte is assumed to be fully dense and impermeable to gases. Thus, thermal insulation and no flux boundary conditions are applied for

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these boundaries. In order to let electrons pass from anode electrode to cathode electrode, the specified voltage condition is applied for the exterior boundaries of the electrodes which faces the wall chambers. Ground voltage is applied for anode side and working voltage is applied for cathode side. As it is assumed that electrolyte is insulated from passing electrons, the insulation condition is applied for electron transport physics at all electrolyte exterior boundaries. Finally, it is important to have proper initial guess for each dependent variable inside all subdomains of computational domain. It can significantly improve the solution convergence. Considering charge (electron and ion) transport physics, a good value for the electric potential in electrodes and its catalyst layers can usually be derived from its boundary condition. For instance, if a boundary has been grounded or set to a cell potential, use that value as the initial value also in the adjacent domain. For the electrolyte potential, a good initial value is often the negative of the equilibrium potential of the grounded electrode. In species transport physics, the inlet molar fraction of species is chosen as initial guess. Thus, the initial guess for molar fraction of hydrogen inside anode gas chamber as well as anode porous layers is equal to 0.95 for this case because the inlet molar fraction of hydrogen is 0.95. The similar condition is governed at cathode side. The initial guess for molar fraction of oxygen inside cathode gas chamber as well as cathode porous layers is equal to 0.21 for this case because the inlet molar fraction of hydrogen is 0.21. In fluid flow physics, the inlet velocity vector and pressure are considered as initial guess within gas chamber. However, since porous electrodes’ resistance to fluid flow is high (this is why the permeability at these layers is quite low as explained in Chapter 6: Fundamentals of fluid flow), the zero-velocity initial guess is opted within these layers. As explained in previous chapters, physics governed in SOFCs are fully coupled. This complicates the solution procedure of the nonlinear differential equations set. To overcome this problem, a proper numerical method such as Finite Control Volume and Finite Element Method (FEM) should be used in order to discrete the differential equations into the algebraic equations. In this case study, FEM is opted. Also, in order to enable the discretization of the geometrical domain into very tiny units of simple shapes, named mesh elements, the meshing process should be performed. FEM reduces the degrees of freedom from infinite to finite with the help of discretization or meshing (nodes and elements). One of the purposes of meshing is to actually make the problem solvable using finite

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element. By meshing, you break up the domain into pieces, each piece representing an element. However, it is important to control the number of mesh elements. Problems with coarser mesh elements may be unstable during solution procedure. On the other hand, problems with finer mesh elements may increase solution time and collapse more memory. Fig. 7.2 shows the meshing structure for case study 1. As can be seen in Fig. 7.2, the finer mesh is opted within catalyst layers where the electrochemical reactions occur and more calculations exist. To have a more comprehensive and realistic prediction of SOFC behavior, all physics including mass, momentum, species, energy, and charge transport should be made into account. Among these physics, momentum transport physics is more complicated and may unstable the solution procedure. This is due to convective term of Navier Stokes equation which is nonlinear. So, it is suggested to solve the set of physics in steps. Initially, solve the nonlinear momentum equations then solutions are stored and applied as initial value for rest of the solution steps. Another solution to capture nonlinearity behavior of momentum equation is to consider second-order element for pressure and velocity field. But this method significantly increases computations and require more RAM. Another nonlinear term that affects the solution procedure is related to electrode kinetics, such as Bulter Volmer kinetics that is described in Chapter 4, Fundamentals of electrochemistry. A good guess for initial value of voltage can improve solver convergence.

Figure 7.2 Mesh structure for case study 1.

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

It is better to use structured meshing. Two mesh types are more common in SOFC modeling. They are mapped and triangular meshes. In this case, 32917 triangular elements are used to solve fully coupled governing equations. The computation time for all scan of working voltages from 1 V to 0.4 V with step 20.1 V is 8 min and 19 s on a personal laptop Corei 7 16 GB RAM. Fig. 7.3 shows the most important plot representing the cell performance which is known as cell performance plot. With the help of this plot we can estimate the amount of maximum power density produced by the cell. This plot can be generated by plotting the power density obtained by crossing the current density produced by cell into the working voltage versus current density. According to this plot, the maximum power density achieved by case study 1 is equal to 1.14015 W/cm2 to our case study 1 which is obtained in 2.2803 A/cm2. Figs. 7.4 and 7.5 show fluid velocity and mixture density along with the axis passing the centerline of the anode gas chamber in different working voltages of 0.8, 0.7, and 0.6 V, respectively. As can be seen, initially fluid velocity has a jump from 0.5 m/s at inlet to 7.1 m/s at 0.5 mm distance from inlet. This is due to no-slip condition applied for chamber walls. This condition forces the flow to rest at walls. According to mass continuity law, the center-velocity has to increase in order to compensate the decrease of the momentum in the vicinity of the wall. This thin layer formed near wall is known as boundary layer that was first introduced by Prandtl in 1904. As the flow progresses, the layer becomes thicker until the velocity profile does not experience any changes. This flow region is 1.20

Wcell (W/cm2)

1.00 0.80 0.60 0.40 0.20 0.00 0

0.5

1

1.5 I (A/cm2)

Figure 7.3 Cell power density plot for case study 1.

2

2.5

3

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Line graph: Velocity field, y component (m/s) 7 6.8 6.6

v (m/s)

6.4 6.2

0.8 0.7 0.6

6 5.8 5.6 5.4 5.2 5 0

0.002

0.004

0.006

0.008

y-coordinate (m)

Figure 7.4 Velocity distribution along with axis passing the centerline of the anode gas channel at 0.8, 0.7, and 0.6 V for case study 1.

called a fully developed flow where the velocity profile remains constant in the flow direction. Thus, in fully developed regime, it is expected that velocity remains constant. However, surprisingly as can be seen in Fig. 7.4, the velocity decreases moderately from 7.1 m/s to about 6.55 m/s. Plus, the slop changes at 0.6 and 0.7 V is slightly higher than 0.8 V. The reason for this behavior of the fluid velocity can be found in Fig. 7.6 where the mixture density distribution along with the same axis is shown. As can be seen, the density distribution behaves different from fluid velocity distribution and it increases with approaching the channel output. This is due to hydrogen consumption and water production occurred in anode catalyst layer. It leads to increased water concentration, more weighted molecule compared to hydrogen weight. By increasing the mixture density, the inertia term in momentum transport equation (left-hand side of Eq. 6.12) is boosted and then the fluid velocity reduces. Figs. 7.6 and 7.7 show fluid velocity and mixture density along with the axis passing the centerline of the cathode gas chamber in different working voltages of 0.8, 0.7, and 0.6 V respectively. As can be seen, the

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

ρmix,a (kg/m3)

0.034

0.8 0.7 0.6

0.033

0.032

0

0.004 y-coordinate (m)

0.008

Figure 7.5 Mixture density distribution along with axis passing the centerline of the anode gas channel at 0.8, 0.7, and 0.6 V for case study 1.

same behavior of the fluid flow at cathode channel inlet is observed; as explained, this due to growth of boundary layers. However, after this point velocity tends to increase slightly and the slop of this change is about more by decreasing the working voltage. This is due to consumption of oxygen which has higher molecular weight compared to the other species, nitrogen, at cathode side. Consequently, the mixture density reduces and then the resistant inertia force is deduced. It is clear that by decreasing the working voltage, the rate of oxygen production increases and then density is reduced, but velocity is increased. Figs. 7.8 and 7.9 depict the molar concentration of hydrogen, oxygen, and water species along with the x-axis crossing middle of the cell at different working voltages of 0.6, 0.7, and 0.8 V, respectively. As expected, hydrogen is consumed within porous anode electrode due to hydrogen reduction equation and also oxygen is consumed, but water is produced within porous cathode electrode due to oxygen oxidation equation. The

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Figure 7.6 Velocity distribution along with axis passing the centerline of the cathode gas channel at 0.8, 0.7, and 0.6 V for case study 1

minimum molar concentration of hydrogen is observed at anode electrolyte interface where the rate of electrochemical reaction is high. Similarly, the minimum oxygen molar concentration and maximum water molar concentration is observed at electrolyte cathode interface as well. It is clear that by decreasing voltage, rate of production and consumption of species should be increased which is relevant from Figs. 7.8 to 7.10. For instance, the maximum hydrogen molar concentration occurs at anode channel which is about 10.6 mol/m3 at 0.8 V. However, its minimum molar concentration occurs at electrode electrolyte interface at 0.6 V and is about 9.25 mol/m3. One of the important criteria in SOFC commercialization is its fuel utilization. In this case study, the average molar fraction of the hydrogen which is used as fuel is plotted versus current density and shown in Fig. 7.11. It can be seen that by increasing the current density, the more amount of fuel is utilized within the cell. Fig. 7.11 shows that the amount for this case is about 15.6% in best situation when 28000 A/m2 cell current density is drawn out from the cell.

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0.328 0.8 0.7 0.6

0.327 0.326 0.325 0.324 0.323 ρmix,c (kg/m3)

0.322 0.321 0.32 0.319 0.318 0.317 0.316 0.315 0.314 0.313 0

0.004 y-coordinate (m)

0.008

Figure 7.7 Mixture density distribution along with axis passing the centerline of the cathode gas channel at 0.8, 0.7, and 0.6 V for case study 1

Fig. 7.12 shows the average molar fraction of oxygen at cathode gas channel outlet versus current density. It is evident that for this case 9.6% of oxygen amount is decreased from the air when 28000 A/m2 cell current density is drawn out from the cell. As stated, one of the SOFC challenges is related to the amount of the maximum temperature occurred within the cell. This high temperature can lead to thermal stresses and shocks. Fig. 7.13 shows the maximum temperature that occurred in the cell versus current density. It is expected that the maximum temperature of the cell increases with rise of the current density. It can be estimated from Fig. 7.13 that the maximum temperature of the cell is 1187.84K in worst working condition. In order to evaluate where this maximum temperature occurs in the cell, plotting the temperature contour as shown in Fig. 7.14 is very useful. As can be seen, the maximum temperature occurs at end edge of the cell adjacent to anode electrode. This is due to convective heat transfer and ohmic heating amount.

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10.6 V = 0.8

10.5

V = 0.7

10.4

V = 0.6

10.3 CH2 (mol/m3)

10.2 10.1 10 9.9 9.8 9.7 9.6 9.5 9.4 9.3 0

0.4

0.8 x-coordinate (m)

1.2

× 10–3

Figure 7.8 Hydrogen molar concentration along with the x-axis crossing middle of the cell at different working voltages of 0.6, 0.7, and 0.8 V for case study 1.

7.2 Case 2: Transient performance analysis of a dual chamber solid oxide fuel cell with hydrogen fuel In this section, the transient performance of the cell described as case study 1 is our aim. So, all physical and geometrical data mentioned in previous section are maintained. Considerable amounts of research work have been conducted on SOFC, making the SOFC close to the commercial applications. One of the challenges for application of SOFCs is their relatively slow response to the input parameters time variation. Understanding the transient behavior of SOFCs is also important for control of stationary utility generators during power system faults, surges, and switching. Therefore, the transient modeling of the fuel cell is useful to predict the cell dynamic behavior and operation. In 1994, Achenbach [28] analyzed the dynamic operation of a planar SOFC. He examined the transient cell voltage performance due to temperature changes and current density with lumped assumption for the cell temperature distribution. Hall and Colclaser [29] also developed a thermodynamic model for

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

CO2 (mol/m3)

2.3

V = 0.8 V = 0.7 V = 0.6

2.2

1.6

1.7

1.8 1.9 x-coordinate (m)

2

× 10–3

Figure 7.9 Oxygen molar concentration along with the x-axis crossing middle of the cell at different working voltages of 0.6, 0.7, and 0.8 V for case study 1.

prediction of transient operation of the tubular SOFC. Sedghisigarchi and Feliachi [30] combined heat transfer dynamics and species dynamics to form a new dynamic model. Also, Xue et al. [31] considered a onedimensional transient model for heat and mass transfer simulation, assuming an electrical circuit includes the ohmic resistances and capacitors for the energy storage mode of operation. Iora et al. [32] considered the internal reforming/shifting reactions in fuel channel in their study. Qi et al. [33,34] developed a quasi 2D model of a tubular SOFC based on the changes along the gas flow direction using the control volume (CV) approach. In their model, the cell length is divided to several serial segments. Each segment includes five CVs, that is, air tube, air channel (in two sections), cell body (electrolyte and electrodes), and fuel channel. They obtained a nonlinear set of differential equations for the heat and mass transfer as well as electrical and electrochemical variables. By solving

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1.7 V = 0.6 V = 0.7

1.6

V = 0.8 1.5

CH2O (mol/m3)

1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0

0.4

0.8 x-coordinate (m)

1.2

× 10–3

Figure 7.10 Water molar concentration along with the x-axis crossing middle of the cell at different working voltages of 0.6, 0.7, and 0.8 V for case study 1.

these equations simultaneously, they calculated the cell time response to the load change. One obvious weakness of these dynamic models is that they used constant heat and mass transfer coefficients based on a fully developed flow approximation at constant wall temperature and mass flux. In 2009, Mollayi et al. [35] presented a 2D transient numerical model to predict the dynamic behavior of a tubular SOFC. In their model, the transient conservation equations (momentum, species, and energy equations) are solved numerically and electrical and electrochemical outputs are calculated with an equivalent electrical circuit for the cell. Their developed model determined the cell electrical and thermal responses to the variation of load current. Also, it predicted the local EMF, state variables (pressure, temperature, and species concentration), and cell performance for different cell load currents. This case study extends the case study 1 results to involve transient behavior of the cell. For this purpose, all data from the previous section

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0.94 0.93 0.92 0.91 0.9 yH2,out (mol/m3)

0.89 0.88 0.87 0.86 0.85 0.84 0.83 0.82 0.81 0

8000

16,000 I (A/m2)

24,000

Figure 7.11 Average hydrogen molar fraction at anode gas channel outlet versus current density for case study 1.

are stored and used as initial values for case study 2. Then, a step change to working voltage is applied in order to observe the transient behavior of the cell. Note that since the transient study is the goal of case study 2, all governing equations including mass, momentum, energy, species, and charge (electron and ion) should be solved considering their time derivative term. For details, the readers are referred to Chapters 4 6. To initiate transient study, two different step changes, including positive and negative in working voltages, are considered as shown in Fig. 7.15. The working voltage varies from 0.7 V to 0.8 V and 0.6 V. The inlet hydrogen molar fraction was maintained at the nominal value of 95%. The step changes in the working voltage were introduced at 10 s. The solution procedure as well as mesh structure for case study 1 and 2 are the same. The solution time for this case is 43 s.

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yO2,out (mol/m3)

0.205

0.2

0.195

0.19 8000

16,000 I (A/m2)

24,000

Figure 7.12 Average oxygen molar fraction at anode gas channel outlet versus current density for case study 1.

Fig. 7.16 shows the response of the cell current density to two different voltage step changes: (1) negative step change of working voltage from 0.7 to 0.6 V and (2) positive step change of working voltage from 0.7 to 0.8 V. Two different trends can be seen including a sudden jump of current density from 1.39 to 1.8 A/cm2 for the case of negative step change and a sudden fall of current density from 1.39 to 1.0142 A/cm2 for the case of positive step change. This is due to the instantaneous electrochemical reaction taking place in the electrodes. After this point, there is a slight decrease from 1.8 to 1.73 A/cm2 for case 1. As a result of this increase in current density, reactant concentration at the reaction layers drops instantaneously. Due to the larger mass transfer time scale from the channel to the electrodes, the reactants at the catalyst layers are not replaced instantly. Consequently, concentration overpotential increases and hence the current density decreases for a

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

1187.84 1177.6 1167.36 1157.12

Tmax (K)

1146.88 1136.64 1126.4 1116.16 1105.92 1095.68 1085.44 1075.2 8000

16,000 I (A/m2)

24,000

Figure 7.13 Maximum temperature of the cell versus current density for case study 1.

period of time which results in an undershoot in the dynamic response of the cell. This can be seen in the insert of Fig. 7.16 which magnifies the initial time intervals after the positive voltage step change. On the other hand, a slight increase from 1.0142 to 1.0186 A/cm2 for case 2 is observed. This is because of decreasing the rate of the electrochemical reactions within catalyst layers, which leads to decreasing the consumption rate of reactants. Thus, concentration overpotential decreases and the current density increases for a while. It results in an overshoot in the dynamic response of the cell. It can be seen that for both cases the current density remains constant after 40 s from applying voltage change. Fig. 7.17 depicts the transient response of the average cell temperature to two different step changes in voltage shown in Fig. 7.15. It can be seen that the average temperature of cell increases from 1095K to 1112K in response to negative step of voltage. This is due to increasing the current density, which leads to increasing the ohmic heat. But decreasing the

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Figure 7.14 Temperature contour at 0.7 V for case study 1.

0.85

Vcell (V)

0.8 0.75

0.7–0.8 V

0.7

0.7–0.6 V

0.65 0.6 0.55 0

100

300

200

400

500

t (s)

Figure 7.15 Different voltage step changes considering for case study 1.

current density and then decreasing the amount of ohmic heat within the cell lead to decrease of cell average temperature. It can be realized that the response of the cell temperature is quite slower than response of current density to step changes of voltage. Because it lasts about 125 s from the initial negative step change of voltage until the cell temperature does

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Figure 7.16 Current density response to (1) a negative step change in voltage from 0.7 to 0.6 V (solid line) and (2) a positive step change in voltage from 0.7 to 0.8 V (dashed line).

not vary with time. This response is a little fast when a positive step change in voltage is applied. As can be seen, the cell temperature decreases from 1095K to 1086K after 110 s which is 15 s faster. Figs. 7.18 and 7.19 show the response of average molar fraction of hydrogen and oxygen at electrode electrolyte interface to step change of voltage, respectively. It is deduced that the step time of the mass diffusion is quite low compared to heat and current density transport phenomena. It causes the hydrogen molar fraction reaching to its new steady-state condition after 20 s for both changes in voltage. The similar behavior is observed in the case of oxygen.

7.3 Case study 3: The effect of coplanar and perpendicular catalyst layer configurations on the performance of a single-chamber SOFC Case study 3 focuses on the numerical evaluation of the effect of coplanar and perpendicular catalyst layer configurations on the performance of a single-chamber SOFC (SC-SOFC). (This section was published in Applied

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Figure 7.17 Temperature response to (1) a negative step change in voltage from 0.7 to 0.6 V (solid line) and (2) a positive step change in voltage from 0.7 to 0.8 V (dashed line).

Thermal Engineering Journal.) Three different catalyst layer configurations, coplanar, single-cell perpendicular, and two-cell stack, are used. Fuel is a mixture of hydrogen and air (50% hydrogen and 50% air by volume). An in-house CFD code is utilized to solve the nonlinear governing equations of mass, momentum, energy, charge balance, and gas-phase species coupled with kinetics equations. As stated, fuel cells directly produce electricity from the external supply of fuel and oxidant [1]. Among different types of fuel cells, SOFCs have gained consideration due to their efficiency and fuel flexibility [11,36 42]. As known, SOFCs operate in the high-temperature range, 700°C 1000°C, and face several problems such as material degradation and gas leakage. The gas leakage is the major obstacle of SOFCs for commercialization [36,38,39]. An alternative to SOFC and its mentioned problems is the use of SC-SOFCs. In SC-SOFC, a mixture of fuel and oxidant are fed directly into the cell [39]. The goal of much research is to improve the performance of the SCSOFCS. To date most research is limited to experimental evaluation of

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Figure 7.18 The average hydrogen molar fraction response to (1) a negative step change in voltage from 0.7 to 0.6 V (solid line) and (2) a positive step change in voltage from 0.7 to 0.8 V (dashed line) at anode electrolyte interface.

the SC-SOFC which only provides information about overall performance of the cell [13,25,26,36,38,41,43 45]. The few existing papers on SC-SOFC numerical models do not provide detailed information about the electrochemical interactions which is needed to improve the performance of SC-SOFC [37,40]. A study by Chung et al. [37] shows that ohmic loss is the major loss in the SC-SOFCs compared to the other two losses (i.e., concentration and activation losses). Their results also reveal that the ohmic loss diminishes by increasing the electrolyte layer thickness. The same results are presented by Akhtar and his team [43]. They reported that increasing the electrolyte layer thickness extend the cross-sectional area available for ionic current flow in lateral direction. Akhtar also evaluated the effect of the cathode to anode distance on gas species as well as the velocity distribution through electrodes. The right-angular configuration for SC-SOFC was initially introduced by Wang et al. [44]. They used methane air mixture as fuel for this configuration. Wang reported that the rightangular configuration exhibit much better performance than coplanar

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Figure 7.19 The average oxygen molar fraction response to (1) a negative step change in voltage from 0.7 to 0.6 V (solid line) and (2) a positive step change in voltage from 0.7 to 0.8 V (dashed line) at cathode electrolyte interface.

configuration. This is due to considerable reduction in the ohmic resistance occurring in right-angular type. They also studied the two-cell right-angular configuration stack performance and found out that the ohmic resistance is the smallest in this configuration. However, their results are limited to experiments and just give information about the overall performance of the cell. The main aim of this case study is to look at the effect of different catalyst layer configurations on the cell performance. The goal is to advice an anode and cathode setup that minimizes the overall losses and increases the SC-SOFC performance. Fig. 7.20 depicts the schematic of the problem. As shown, a mixture of hydrogen and air (50% hydrogen, 50% air by volume) is used to ensure the system safety. The hydrogen is sufficiently diluted with nitrogen to avoid explosion in the system [22]. As shown in Fig. 7.20 the positive electrode, electrolyte, and the negative electrode (PEN) is placed at the middle of the channel. Three different configurations are selected: (1) coplanar configuration (case 1) in which two electrodes are placed on the same side of the electrolyte, (2) perpendicular configuration (case 2) in

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Figure 7.20 The schematic of the problem: (A) the whole computational domain with the inlet and the outlet (B) different catalyst layer configurations considered in modeling.

which electrodes are placed on two mutually perpendicular planes, and (3) a two-cell stack in which two single cells are placed on the electrolyte layer to form a simple stack (case 3). The cell consists of five layers: (1) anode made of nickel (Ni), (2) anode catalyst layer made of Ni-YSZ, (3) electrolyte made of YSZ, (4) cathode catalyst layer made of YSZ-LSM, (5) and cathode layer made of LSM. Table 7.4 shows the geometrical parameters. The fluid is assumed to behave as an ideal gas and the flow is steady, two-dimensional, compressible, and laminar. All fluid properties vary with temperature. The electrolyte is fully impermeable and the electrodes are selective, which means electrochemical reactions described in Chapter 1, Introduction to fuel cells, occur in the electrodes. The ohmic resistance due to electron transport and thermal diffusion is neglected. This is due to the electronic conductivity of the electrodes that is noticeably higher than the ionic conductivity. The inertia term (Stokes Brinkman’s assumption) is also neglected. An in-house CFD code based on a FEM was developed and utilized. The code uses triangular meshes. A set of equations are solved in steps. Initially, the code solves the charge conservation coupled with the energy equation. Then it solves the nonlinear momentum equations and terminates when the Stefan Maxwell equation is solved. To improve the accuracy of the code, the second-order elements for the velocity

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Table 7.4 Geometrical data for case study 3. Description

Symbol

Value

Channel width Channel height Anode thickness Anode catalyst layer thickness Electrolyte thickness Cathode thickness Cathode catalyst layer thickness Electrolyte width Electrode width x-value of electrolyte to anode distance (edge to edge)

Wch Hch ta tac te tc tcc We Welec d

150 (mm) 25 (mm) 70 (µm) 5 (µm) 3 (mm) 50 (µm) 5 (µm) 10 (mm) 2 (mm) 1.5 (mm)

components and species mass fraction are opted. For other dependent variables, linear elements are used. The relative tolerance is set to 1 3 1026. A comparison study between different configurations of a SC-SOFC is performed and reported. Note that in all cases the anode distance from the electrolyte edge (the “d” parameter shown in Fig. 7.20) remained constant. All input parameters that are tabulated in Table 7.3 except the inlet mass fractions and temperature are fixed. At the inlet, the temperature is 750°C and the mixture consists of 50% hydrogen, 10.5% oxygen, and 30.5% nitrogen by volume. Figs. 7.21 and 7.22 show the polarization and I P curves for case 1, case 2, and case 3, respectively. As shown in the figures, case 2 shows the better performance compared to case 1. The maximum power density produced by case 2 is 20.9 mW/cm2, whereas case 1 produces only 12.5 mW/cm2 in its best condition. In addition, a two-cell stack (i.e., case 3) produces near a double performance. Fig. 7.23 depicts the oxygen ions path for case 1, case 2, and case 3 that takes place from cathode to anode. As shown, the oxygen ion path of case 2 is noticeably shorter than case 1 and therefore a better performance is achieved. In addition, there is no ion transmission at the right-hand side of the electrolyte domain of case 2 (see Fig. 7.23). Adding another anode and cathode electrode on the other face of the electrolyte (i.e., case 3) improves the cell performance and overcomes this problem. The maximum power density produced by case 3 is 43.4 mW/cm2. Fig. 7.24 depicts the normal ionic current density distribution of both cases along the anode catalyst layers. It is evident that there are much more oxygen ions at the edge of the anode where oxygen ions have to

166

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

0.95 Case1 0.85

Case2

Cell voltage (V)

Case3 0.75

0.65

0.55

0.45

0.35 0

20

40

60

80

100

120

Current density (mA/cm2)

Figure 7.21 Polarization curve for cases 1, 2, and 3.

50 45

Power density (mW/cm2)

40 35 Case1 30 Case2 25

Case3

20 15 10 5 0 0

20

40

60

80

100

Current density (mA/cm2)

Figure 7.22 Power density versus current density for cases 1, 2, and 3.

120

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167

Figure 7.23 Ionic current density stream lines.

pass the shortest path from cathode to anode for all cases. However, the ionic current density produced by case 3 is greater. In case 3, the normal ionic current density at the edge of the anode that faces the cathode is slightly greater for both anodes. Figs. 7.25 and 7.26 show the hydrogen and the oxygen molar concentration distribution of all cases through the axis across the middle of anode and cathode catalyst layer. In these locations, the hydrogen and oxygen are consumed and produce electrons and ions, respectively. However, hydrogen consumption varies according to its distance from the edge of the electrode position. Case 2 reveals better hydrogen consumption. This is due to an increase in the hydrogen molecules’ participation in the electrochemical reactions for case 2 compared to case 1. The surprising phenomenon happening in both cases is that a slight increase of hydrogen molar concentration occurs at the cathode side. This is because hydrogen compared to oxygen diffuses easier due to its small size. The increase in hydrogen concentration reduces the number of oxygen molecules at the

168

Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Figure 7.24 Normal ionic current density distribution along anode catalyst layer.

Figure 7.25 Hydrogen molar concentration distribution along axis of interest, case 1.

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Figure 7.26 Oxygen molar concentration distribution along cathode catalyst layer.

cathode side by electrochemical reactions and causes the performance to decrease. Similarly, case 2 reveals better oxygen consumption and more oxygen participation in the electrochemical reaction compared to case 1. Furthermore, the lowest level of oxygen concentration occurs where the highest level of ionic concentration is observed. It was shown that stacking a perpendicular type of SC-SOFC improves SC-SOFC performance. So, it is necessary to perform a parametric study of a two-cell stack in order to get the best sketch of case 3 showing the best performance. The “d” symbol as shown in Fig. 7.20 is considered as a parameter. Six different values are opted: 0 mm, 0.5 mm, 1 mm, 1.5 mm, 2 mm, and 2.5 mm. The polarization and I P curves as a function of “d” are illustrated in Figs. 7.27 and 7.28, respectively. It is clear from Fig. 7.27 that total overpotential occurred within the cell increases by growth of “d” value since increasing the cathode to anode distance of each cell enhances the amount of ohmic overpotential. However, increasing the “d” value from 0 to 0.5 mm does not affect the performance of the cell that much. As shown in Fig. 7.27, the best performance is observed when the “d” value is set to 0 and 0.5 mm. In this case, the maximum power density is obtained as 484.6 and 472.6 W/m2, respectively. Furthermore, the deviation between different “d” values is more evident at high range of current density (approximately above 400 A/m2).

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Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells

Figure 7.27 Polarization curves for different “d” values of case 3 shown in Fig. 7.20.

Figure 7.28 I P curves for different “d” values of case 3 shown in Fig. 7.20.

According to case study 3, performance analysis of two different SC-SOFC catalyst layer configurations is reported. Results show about 44% increase in performance for the perpendicular configuration, case 2, compared to the coplanar configuration, case 1. The maximum power density for case 2 is estimated as 12.525 mW/cm2 and for case 1 about 21.8 mW/cm2. It is shown that the oxygen ions transportation path, from cathode to anode, play a key factor in performance enhancement of case 2. Furthermore, the bulk of hydrogen is formed at the cathode side in both cases which can cause oxygen molecules to have difficulty in reaching the cathode catalyst layer and lead to poor performance. Therefore, hydrogen bulk formation at the cathode side is a problem for the single-chamber configuration that should be considered.

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Index Note: Page numbers followed by “f” and “t” refer to figures and tables, respectively.

A Afterburners, 55 All porous solid oxide fuel cell (APSOFC), 39 40 Anode-supported solid oxide fuel cell, 30 32, 33f, 33t Axial symmetry, 138 139 boundary condition, 98

B Balance of plant (BoP), 48 Blackbody surfaces, 109 110 Body (volume) force, 134 “Boundary layer,”, 108 Boyle’s law, 77 78 Brayton Rankine cycle, 49, 51 Brayton (gas) regenerative cycle, 49 51, 49f Brinkman equation, 132 133 Bulter Volmer formula, 129

C Carbonate ion exchange fuel cell, 7 8, 8f Case studies The effect of coplanar and perpendicular catalyst layer configurations on the performance of a single-chamber SOFC, 160 170 Stationary performance analysis of a dual chamber solid oxide fuel cell with hydrogen fuel, 142 152 Transient performance analysis of a dual chamber solid oxide fuel cell with hydrogen fuel, 153 160 Catalyst layer, 75 Cathode-supported solid oxide fuel cell, 28 30, 31f CFD. See Computational fluid dynamics (CFD) Chapman Enskog formula, 84 Charles law, 78

COMB. See Combustor (COMB) Combustion engines, 1 2 Combustor (COMB), 58 Computational fluid dynamics (CFD), 141 142 Conduction heat transfer, 103 107 Conservation of species, 80 87 Continuity boundary condition, 98, 121 Control volume (CV), 153 155 Convection heat transfer, 107 109 Convective heat transfer coefficient, 108 Conventional electric power-producing devices, 1 2 Conventional interconnector, 34 35 Conventional SOFCs, 18 CV. See Control volume (CV)

D Darcy’s law, 132 Darcy velocity, 132 DF-SOFC. See Direct-flame SOFC (DFSOFC) DGM. See Dusty-gas model (DGM) Diffuse surfaces, 109 110 Direct-flame SOFC (DF-SOFC), 18, 41 Direct integration, 56 57 Doped LaCrO3 perovskite, 19 Doped LaMnO3, 19 Double-pipe heat exchanger, 71 Dry reforming, 128 Dual chamber SOFC, 18, 36 37 stationary performance analysis of, 142 152, 143f transient performance analysis of, 153 160 Dusty-gas model (DGM), 82

E Ejectors, 53 54, 54f Electrical potential boundary condition, 97 98

175

176

Index

“Electric shaft prolongation,”, 55 Electrochemical reaction rate, 93 96 Electrochemistry conservation of species, 80 87 gas mixture concepts ideal gas mixtures, 77 78 mass fractions and mole fractions, 76 77 properties of gas mixtures, 79 80 Electrode current collector, 97 Electrode electrolyte interface, 75 “Electrolyte,”, 6 Electrolyte-supported solid oxide fuel cell, 28, 29f Emissivity, 109 110 Equilibrium statistical mechanics, 77 78 Exergo-economic analysis, 48

F Fick’s law of diffusion, 82 Finite Element Method (FEM), 146 147 Flat plate, 21 22 Flow rate, 129 Fluid flow boundary conditions axial symmetry, 138 139 continuity, 139 inlet boundary condition, 134 135 outlet boundary condition, 136 137 wall boundary condition, 137 138 conservation of linear momentum Body (volume) force, 134 Brinkman equation, 132 133 Navier Stokes equations, 133 conservation of mass law of conservation, 126 131 mass sources caused by chemical reactions, 127 129 mass sources caused by electrochemical reactions, 129 131 Free enthalpy, 8 9 Fuel cell description of, 1 2 first hydrogen fuel cell, 2f operation of, 2 3 schematic illustration of, 3f thermodynamics of, 8 14

types of carbonate ion exchange fuel cell, 7 8, 8f hydroxide ion exchange fuel cell, 3 4, 4f oxide ion exchange fuel cell, 4 5, 5f proton exchange fuel cell, 6 7, 6f Fuel desulfurization, 51 52 Fuel-rich condition, 37 Fully developed flow, 148 149 Fully porous SC-SOFC, 38

G Gas mixture concepts, 76 80 properties of, 79 80 Geometrical types, SOFC button design, 25 26, 26f, 27f delta design, 24, 25f high-power density design, 23 24, 24f planar design, 21 22, 21f tubular design, 22 23, 22f Gibbs free energy, 8 9 Gray surfaces, 109 110 Green energy, 1

H Heat equation in channels, 117 in electrolytes, 115 116 in porous electrodes, 116 117 Heat exchangers, 52 53, 59, 65 Heat recovery steam generator (HRSG), 59 Heat transfer, fundamental of boundary conditions for solid oxide fuel cells continuity, 121 outflow, 121 122 specified heat flux, 121 specified temperature, 120 surface-to-ambient radiation, 123 124 symmetry, 122 thermal insulated boundary, 120 different modes of heat transfer, 103 113

Index

conduction heat transfer, 103 107 convection heat transfer, 107 109 radiation heat transfer, 109 113 Rosseland approximation, 112 113 Schuster Schwartzchild two-flux approximation, 111 112 energy conservation heat equation in channels, 117 heat equation in electrolytes, 115 116 heat equation in porous electrodes, 116 117 solid oxide fuel cell’s source terms heat source generated by chemical reactions, 119 irreversible heat source, 118 Joule or Ohmic heat source, 118 reversible heat sources, 118 119 High-power density (HPD), 23 HPD. See High-power density (HPD) HPD 5, 23 24 HPD 10, 23 24 HRSG. See Heat recovery steam generator (HRSG) Hybrid systems, SOFC balance of plant equipment afterburners, 55 ejectors, 53 54, 54f fuel desulfurization, 51 52 heat exchangers, 52 53 other components, 56 power electronics, 55 56 reformer, 54 55 different configurations of direct thermal coupling scheme, 58 61, 58f, 60f, 61f indirect thermal coupling scheme, 61 63, 62f, 63f other types of coupling, 63 65 exergy-based methods, 47 48 mathematical modeling, 65 73 SOFC and internal combustion engine, 47 48 solid oxide fuel cell/gas turbine hybrid cycle, 56, 57f strategies for improving the efficiency of, 48 49

177

thermodynamic cycle, 49 51 Hydrosulfide (H2S), 51 52 Hydroxide ion exchange fuel cell, 3 4, 4f

I Ideal gas, 77 78, 126 127 mixtures, 77 78 Indirect integration, 57 Indirect internal reformers, 54 55 Inflow boundary conditions, 96 Inlet boundary condition, 134 135 Insulation boundary conditions, 97 Integrated coal gasification and SOFC/ GT/ST hybrid system, 64f Intermediate-temperature SOFCs (ITSOFCs), 27 28 “Internal reforming,”, 7 Irreversible heat source, 118 Ion transport, 93 ITSOFCs. See Intermediate-temperature SOFCs (ITSOFCs)

J Joule or Ohmic heat source, 118

K Knudsen diffusion, 85 86

M Mass continuity law, 148 149 Mass fraction, 76 Mass transport, 82, 85 86, 98 Mathematical modeling, 65 73 “Membrane,”, 3 Methane-reforming reaction, 89 Methane steam reforming reaction (MSR), 128 Molar concentration diffusion flux, 82 Molar diffusional flux model, 82 Mole fraction, 76 77 MSR. See Methane steam reforming reaction (MSR)

N Navier Stokes equations, 133 Newton’s Law of Cooling, 108 No-chamber SOFC, 41

178

Index

No-chamber solid oxide fuel cell, 41 42

O OCV. See Open circuit voltage (OCV) Ohmic heating, 118 Open circuit voltage (OCV), 25 26 Outflow boundary condition, 96 97, 121 122 Outlet boundary condition, 136 137 Oxide ion exchange fuel cell, 4 5, 5f

P PAFC. See Phosphoric acid fuel cell (PAFC) PEFC. See Polymer electrolyte fuel cell (PEFC) PEMFCs. See Proton exchange membrane fuel cells (PEMFCs) Phosphoric acid fuel cell (PAFC), 6 Polymer electrolyte fuel cell (PEFC), 6 Polytropic process, 67 68 Positive-electrode/electrolyte/negativeelectrode (PEN), 21 22, 25 Power electronics, 55 56 Proton exchange fuel cell, 6 7, 6f Proton exchange membrane fuel cells (PEMFCs), 3, 6, 64

R Radiation heat transfer, 109 113 Rankine (steam) cycle, 49, 49f, 51 “Rate-determining step,”, 93 94 Recuperators, 52 Reformer, 54 55 Reversible heat source, 118 119 Reversible processes, thermodynamics of, 13 Rosseland approximation, 112 113

S SCCM. See Standard cubic centimeter per minute (SCCM) Schuster Schwartzchild two-flux approximation, 111 112 SC-SOFCs. See Single-chamber solid oxide fuel cell (SC-SOFCs)

Single-chamber solid oxide fuel cell (SCSOFCs), 37 40, 103, 160 161 coplanar and perpendicular catalyst layer configurations, 160 170 Small-stage/infinitesimal-stage efficiency, 68 SOFC. See Solid oxide fuel cell (SOFC) SOFC and PEMFC, integration of, 64f Solid oxide fuel cell (SOFC), 5, 47, 75, 125, 141 142 boundary conditions for axial symmetry boundary condition, 98 continuity boundary condition, 98 electrical potential boundary condition, 97 98 inflow boundary conditions, 96 insulation boundary conditions, 97 outflow boundary condition, 96 97 cell types in terms of its chamber number dual-chamber solid oxide fuel cell, 36 37 no-chamber solid oxide fuel cell, 41 42 single-chamber solid oxide fuel cell, 37 40 cell types in terms of its support anode-supported solid oxide fuel cell, 30 32, 33f, 33t cathode-supported solid oxide fuel cell, 28 30, 31f electrolyte-supported solid oxide fuel cell, 28, 29f classification based on flow patterns, 32 35 geometrical types button design, 25 26, 26f, 27f delta design, 24, 25f high-power density design, 23 24, 24f planar design, 21 22, 21f tubular design, 22 23, 22f history, 17 21 single and stack cell designs, 42 45 species source terms in chemical reactions, 88 91 electrochemical reaction rate, 93 96 electrochemical reactions, 91 96 Species continuity equation, 81 Specified heat flux, 121

Index

179

Specified temperature, boundary conditions, 120 Sr- and Mg doped LaGaO3 (LSGM), 20 Stacking, 42 43 Standard cubic centimeter per minute (SCCM), 134 135 Standard temperature and pressure (STP), 9 Statistical thermodynamics, 77 78 Steam-to-carbon ratio, 54 Stefan Boltzmann law, 109 110 Stefan Maxwell model, 82 83 STP. See Standard temperature and pressure (STP) Sulfur-free natural gas, 51 Surface-to-ambient radiation, 123 124 Symmetry, boundary condition, 122

Thermal insulated boundary, 120 Thermal radiation, 109 110 Thermodynamic cycle, 49 51 TIT. See Turbine inlet temperature (TIT) T-s plots. See Temperature-entropy (T-s) plots Turbine inlet temperature (TIT), 49 50 Two-dimensional model, 141 142

T

YSZ. See Yttria-stabilized zirconia (YSZ) Yttria-stabilized zirconia (YSZ), 111 113, 142 143

Temperature-entropy (T-s) plots, 49 Thermal conductivity, 104 105

V Volume force, 134

W Wall boundary condition, 137 138 Water gas shift reaction (WGSR), 89, 128

Y